You may not trust the idea of original sin being inherited, but Methodists absolutely believe in the doctrine of original sin being passed on from Adam and Eve to every human. As a biblical doctrine, original sin is an essential part of the foundation of Christianity, and John Wesley defended it through reason and experience. Wesley concludes this sermon by saying:
Know your disease! Know your cure! Ye were born in sin: Therefore, 'ye must be born again,' born of God. By nature ye are wholly corrupted. By grace ye shall be wholly renewed. In Adam ye all died: In the second Adam, in Christ, ye all are made alive. 'You that were dead in sins hath he quickened:' He hath already given you a principle of life, even faith in him who loved you and gave himself for you! Now, 'go on from faith to faith,' until your whole sickness be healed; and all that 'mind be in you which was also in Christ Jesus!'
Apart from God’s grace leading us to redemption, we have no hope of deliverance from the control of original sin over us. By God’s grace, made available to all in Jesus Christ and through discipleship to him with his church, we do. Source: https://www.umc.org/en/content/ask-the-umc-how-did-john-wesley-understand-original-sin
Given that Methodists do not believe that children are bon innocent, then the answer to your question, ‘is it possible that one can remain sinless till they die?’ must be no. As to the question, ‘how do Methodists deal with this?’ we need to dig a little deeper to understand that, with God’s grace, it is possible to aspire to holiness and to seek perfection. Personally, I don’t think any of us will attain perfection this side of heaven, but that’s no excuse not to try!
This article http://ee.umc.org/what-we-believe/does-the-united-methodist-church-believe-that-babies-are-born-in-sin explains that although Methodists do not choose to follow the way of sin, the nature of man is inclined to evil. The article goes on to explore Romans chapter 5, which is all about salvation. Here is a brief extract:
"as by one man's disobedience [Adam's] the many [meaning all who are born] were made sinners, so by the one man's obedience [Jesus] the many will be made righteous." This is Paul's way of spelling out both the doctrine of sin and the doctrine of salvation. Remember here, we are dealing with Paul's way of setting this up. Christ can redeem all because his faithfulness to God in perfect love and obedience matches and exceeds the disobedience of one man, Adam.
The notion of original sin does not compute very well with the modern outlook. Most of the 20th century church tried to dance around it and then wondered why Jesus' saving work was hard to get serious about. "If we say we have no sin, we deceive ourselves and the truth is not in us." says 1st John, vs. 8.
The point is that we, from birth, need the grace of God available in Jesus Christ. We cannot hope in some tiny spark of goodness at our core that is always there to get us through. We are without merit or claim upon God on our own. This is a hard pill to swallow in our "enlightened" and modern perspective. On the other hand, what a gracious hope and gospel we proclaim and live if we simply accept the desperate need we are in from the beginning and the washing of water and the word in baptism where God claims us as God's own in union with Christ, dying to sin and living alive to God by the power of the Spirit.
Methodists deal with the issue of inherited sin by acknowledging it is only by the grace of God that we can die to sin and be spiritually reborn by the power of the Holy Spirit that indwells believers.