Apologize for the length, the question is so good that I am answering not just for you but digging up worthy references for myself.
A good place to start for an evangelical answer is with two evangelical theologians famous for having an acute sense of the sinfulness of man and the nature of God's grace in the Christian. John Owen and Jonathan Edwards both wrote extensively on the subject.
The main idea expressed by both of these writers is that a Christian is free from sin because in Christ they are translated from the rule of Law to the rule of Grace. Freedom from sin ultimately means freedom from Law because it is God's holy law that condemned man into a totally depraved sinful nature. Unless this curse of the Law for Adam's sin is removed in Christ's death, on whom our sin is made to be Christ's, so that he both bears the guilt and penal wrath of God for it, then we are not free. Therefore this freedom primarily means free from the consequences of sin from the legal standpoint under God's holy law.
However this freedom from the Law necessarily means new birth into the rule of grace, which means that inwardly the Christian is no longer under the dominion of law and sin but under grace leading to a more and more holy life. This 'holy life' is considered the life of the Spirit as contrasted with the life of the 'flesh' and although the Christian has this dual nature causing intense struggle with sin, the spiritual life is said to 'have dominion' over the flesh.
It is this subject of 'dominion' where evangelicals do not all agree, the split primarily resting between Calvinistic views and Arminian views of perfection. I will focus on Calvinistic views in my post as I consider Arminian views on this subject as a weakness of faith. As far as the final state in heaven, of course all Christians believe it is absolute freedom forever, so the question amounts to how much sin can we still have in our life and yet still claim to have faith in God and still claim that grace has dominion in our life. Put another way, 'How much freedom from sin can we experience on this side of heaven?' Or again, 'What does it mean that we are no longer under the dominion of sin but under the dominion of grace, when we have so much sin still in our lives?'
First let's look at typical thoughts by John Owen in his TREATISE
of THE DOMINION OF SIN AND GRACE; wherein sin’s reign is discovered, in whom it is, and in whom it is not; how the law supports it; how grace delivers from it, by setting up its dominion in the heart.
This treatise is based on:
For sin shall no longer be your master, because you are not under the law, but under grace. (Romans 6:14, NIV)
First we should take note that the Christian struggle with sin is not a proper cause for our faith to shrink but for it to take consolation for the fact that we have two natures warring with one another. This confirms we are a Christian and means we have the Spirit. The Spirit, accordion to Ephesians Ch 1 is a 'seal'. this seal that both proves we are God's children and is a down payment, or first fruit, guaranteeing the whole future glory of heaven.
Under this comforting aspect, Owen also provides similar comforting words:
Where men are engaged in a constant conflict against sin; where they look upon it and judge it their chiefest enemy, which contends with them for their souls and their eternal ruin; where they have experience of its power and deceit, and through the efficacy of them have been often shaken in their peace and comfort; where they have been ready to despond, and say they shall one day perish under their powers,—it is a gospel word, a word of good tidings, that gives them assurance that it shall never have dominion over them. The ground of this assurance is, that believers are “not under the law, but under grace.” (John Owen's Works, Vol 7, P542)
But to now directly answer the question. John Owen also asks:
But the inquiry hereon is, how it follows from hence that sin shall not have dominion over us, that sin cannot extend its territories and rule into that state, and in what sense this is affirmed. (P545)
This, then, is the present case supposed and determined by the apostle: “You that are believers are all of you conflicting with sin. You find it always restless and disquieting, sometimes strong and powerful When it is in conjunction with any urgent temptation, you are afraid it will utterly prevail over you, to the ruin of your souls. Hence you are wearied with it, groan under it, and cry out for deliverance from it.” All these things the apostle at large insists on in this and the next chapter. “But now,” saith he, “be of good comfort; notwithstanding all these things, and all your fears upon them, sin shall not prevail, it shall not have the dominion, it shall never ruin your souls.” But what ground have we for this hope? what assurance of this success? “This you have,” saith the apostle, “ ‘Ye are not under the law, but under grace;’ or the rule of the grace of God in Christ Jesus, administered in the gospel.” But how doth this give relief? “Why, it is the ordinance, the instrument of God, which he will use unto this end,—namely, the communication of such supplies of grace and spiritual strength as shall eternally defeat the dominion of sin.” (P546)
Fine, we have said that a Christian has many sins, but they do not have dominion over him. In other words that is he is not really enslaved to sin as he once was before. What does this mean? A common confusion here is to think in terms of percentages. A Christian is (some may think) someone whose heart is 80-90 percent pure, or at least more than fifty percent pure, otherwise he can never say Jesus is his 'Lord'. Some may actually imagine that a Christian can be 100% without sin for maybe even a whole day, but this imagination is quite contrary to the truth of every Christian experience and this type of thinking is not held at all by great evangelical theologians such as John Owen, or Jonathan Edwards. In fact men who really know themselves would never claim to love God even one half as much as they ought to at any given moment, not even for a second.
Here is a good place to watch how Edwards handles the concept in his work on Original Sin. First he argues that even though a Christian is not under the dominion or Lordship of sin, the best of Christians have more of their heart still in a sinful state than a holy one:
The sum of our duty to God, required in his law, is love to God; taking love in a large sense, for the true regard of our hearts to God, implying esteem, honor, benevolence, gratitude, complacency, etc. ...It therefore appears from the premises, that whosoever withholds more of that love or respect of heart from God which his law requires, than he affords, has more sin than righteousness. Not only he that has less divine love, than passions and affections which are opposite; but also he that don't love God half so much as he ought, or has reason to do, has justly more wrong than right, imputed to him, according to the law of God, and the law of reason; he has more irregularity than rectitude, with regard to the law of love. The sinful disrespect or unrespectfulness of his heart to God, is greater than his respect to him.
But what considerate person is there, even among the more virtuous part of mankind, but what would be ashamed to say, and profess before God or men, that he loves God half so much as he ought to do; or that he exercises one half of that esteem, honor and gratitude towards God, which would be altogether becoming him; considering what God is, and what great manifestations he has made of his transcendent excellency and goodness, and what benefits he receives from him? And if few or none of the best of men can with reason and truth make even such a profession, how far from it must the generality of mankind be? (Jonathan Edwards on Original Sin, Page 140)
However when we look at Romans 8 we must feel that there needs some explanation to be made in Edward's assertion for how can the majority of our heart still be under sin, when we are no longer under its dominion? This seems so foolish to the carnal mind! For example what about these verses:
1 Therefore, there is now no condemnation for those who are in Christ Jesus, 2 because through Christ Jesus the law of the Spirit who gives life has set you free from the law of sin and death. 3 For what the law was powerless to do because it was weakened by the flesh, God did by sending his own Son in the likeness of sinful flesh to be a sin offering. And so he condemned sin in the flesh, 4 in order that the righteous requirement of the law might be fully met in us, who do not live according to the flesh but according to the Spirit. (Romans 8:1-4, NIV)
Therefore, brothers and sisters, we have an obligation—but it is not to the flesh, to live according to it. 13 For if you live according to the flesh, you will die; but if by the Spirit you put to death the misdeeds of the body, you will live.
14 For those who are led by the Spirit of God are the children of God. (Romans 8:12-14, NIV)
Edwards get very much into fine detail in his own answer: (I quote him in entirety for my own future reference as it was hard for me to find again and is really a rare gem that expresses the depths of authentic evangelical faith with unusual clarity of mind).
The Objection is this: That the argument seems to prove too much, that it will prove, that even good men themselves have more sin than holiness; which also has been supposed. But if this were true, it would follow, that sin is the prevalent principle even in good men, and that it is the principle which has the predominancy in the heart and practice of the truly pious; which is plainly contrary to the Word of God.
I answer, If it be indeed so, that there is more sin, consisting in defect of required holiness, than there is of holiness in good men in this world; yet it will not follow, that sin has the chief government of their heart and practice, for two reasons.
They may love God more than other things and yet there may not be so much love, as there is want of due love; or in other words, they may love God more than the world, and therefore the love of God may be predominant, and yet may not love God near half so much as they ought to do. This need not be esteemed a paradox: a person may love a father, or some great friend and benefactor, of a very excellent character, more than some other object, a thousand times less worthy of his esteem and affection, and yet love him ten times less than he ought; and so be chargeable, all things considered, with a deficiency in respect and gratitude, that is very unbecoming and hateful. If love to God prevails above the love of other things, then virtue will prevail above evil affections, or positive principles of sin; by which principles it is, that sin has a positive power and influence. For evil affections radically consist in inordinate love to other things besides God. And therefore, virtue prevailing beyond these, will have the governing influence. The predominancy of the love of God in the hearts of good men is more from the nature of the object loved, and the nature of the principle of true love, than the degree of the principle. The object is one of supreme loveliness; immensely above all other objects in worthiness of regard; and 'tis by such a transcendent excellency, that he is God, and worthy to be regarded and adored as God, and he that truly loves God, loves him as God. True love acknowledges him to be God, or to be divinely and supremely excellent, and must arise from some knowledge, sense and conviction of his worthiness of supreme respect. And though the sense and view of it may be very imperfect, and the love that arises from it in like manner imperfect; yet if there be any realizing view of such divine excellency, it must cause the heart to respect God above all.
Another reason, why a principle of holiness maintains the dominion in the hearts of good men, is the nature of the Covenant of Grace, and the promises of that covenant, on which true Christian virtue relies, and which engage God's strength and assistance to be on its side, and to help it against its enemy, that it may not be overcome. The just live by faith. Holiness in the Christian, or his spiritual life, is maintained, as it has respect by faith to its Author and Finisher, and derives strength and efficacy from the divine Fountain, and by this means overcomes. For, as the Apostle says, "This is the victory that overcomes the world, even our faith" [I John 5:4]. 'Tis our faith in him who has promised, never to leave nor forsake his people, and not to forsake the work of his own hands, nor suffer his people to be tempted above their ability, and that his grace shall be sufficient for them, and that his strength shall be made perfect in weakness, and that where he has begun a good work he will carry it on to the day of Christ. (Jonathan Edwards on Original Sin, Page 145-146)
Conclusion: Only as Christians are we really aware of our many sins and have a mournful daily struggle against them, even to the occasional doubting of the very genuineness of our own faith. However as the Spirit has been given to us to give us comfort and assurance of our salvation, we, bumbling along in our many failures, are aware that we are actually being swept along powers of grace along an unbreakable mighty rule of Christ as our high priest, ensuring our absolute certain arrival into glory where we will forever be free of all effects of sin forever. Even as we doubt this grace as it is 'too good to be true' we are being formed into objects of grace meant to reveal this gracious quality of God to the praise of its excellence, which is the entire purpose of life and salvation. Death is no longer a fearful unknown answering our condemned conscience for the punishment of sin. Rather our daily struggle is a hopeful one, looking at the grave as a curtain behind which everlasting joys are waiting, even as we doubt it. Even in the midst of our doubts of God's grace, his grace will be a dominion that will overcome into everlasting praise.
Note: This is the second time I wrote this answer as just after my first version someone pickpocketed my iPhone which had my un-posted answer. Needless to say I was all the more determined to re-post this a few hours later as I was not willing that the Devil so easily take away the fruit of my effort ;)