I have found that the first part of Hebrews 10 addresses this question rather directly. We are all familiar with the Jewish sacrifices of the Old Testament and we understand that they were intended as a foreshadowing of the great and last sacrifice of the Savior. Here are the verses.
1 For the law having a shadow of good things to come, and not the very image of the things, can never with those sacrifices which they offered year by year continually make the comers thereunto perfect. 2 For then would they not have ceased to be offered? because that the worshippers once purged should have had no more conscience of sins. 3 But in those sacrifices there is a remembrance again made of sins every year. 4 For it is not possible that the blood of bulls and of goats should take away sins. 5 Wherefore when he cometh into the world, he saith, Sacrifice and offering thou wouldest not, but a body hast thou prepared me: 6 In burnt offerings and sacrifices for sin thou hast had no pleasure. 7 Then said I, Lo, I come (in the volume of the book it is written of me,) to do thy will, O God. 10 By the which will we are sanctified through the offering of the body of Jesus Christ once for all.
So in this you are correct that there is no forgiveness from God without sacrifice, but that sacrifice is Jesus who is the Christ. After the which, He asks of us a much smaller sacrifice; that we endeavor to repent of our sins.
21 You have no part or portion in this matter, for your heart is not right before God. 22 Therefore repent of this wickedness of yours, and pray the Lord that, if possible, the intention of your heart may be forgiven you. Acts 8:21-22
And this often repeated verse,
Take My yoke upon you and learn from Me, for I am gentle and humble in heart, and you will find rest for your souls. 30"For My yoke is easy and My burden is light." Matthew 11:29-30
I think most people read "My yoke is easy and my burden is light" and hear "My yoke and my burden are nonexistent". We still have a responsibility to participate in the process but the Savior has, by virtue of His sacrifice, put it within our reach. The personal sacrifice that he asks of us is,
The sacrifices of God are a broken spirit; A broken and a contrite heart, O God. Psalm 51:17
That sounds much easier, that's a far lighter burden for me to carry than to be expected to carry the full weight of my sins. Such a weight would necessarily crush me and condemn me to hell.
So I would agree with you that sacrifice is required before forgiveness from God can be given. The required sacrifice was paid in full by Jesus giving Him the right to dole out forgiveness to any who are willing to pay the pittance that He asks for, a broken heart and a contrite spirit.