I know it was Adam and Eve's sin-but why does their sin have to affect us. Doesn't God give free will? Why do we have to live with the mistake of them?
You can't understand the fault behind sin if you first don't understand what sin is. I see you're quoting basic Bible teachings, meaning that in the Bible sin is the transgression of the law (found in the first five books of the Bible), similarly with criminal law you break a law and receive punishment, therefore you are responsible for your actions and whatever Adam and eve did is their problem, but the Bible directly quotes that the punishment of sin is death and that is your problem.
It is your fault because if you know you shouldn't do it then that is your fault. Its not like you created sin but you still should try very hard not to. Go to google and search "What does the bible say about sin?". It will pull up many verses and you should be able to further ingage in it and learn about Jesus Christ our savior.
Sin, singular, is the absence of spirit. Adam had a spirit but it died and so he was unable to bequeath a spirit to his offspring. That is why everybody is born in sin, not because Mommy and Daddy did the big nasty. That is why a savior had to die in our place and make it possible to create a new spirit for us.
Sins, plural, are mistakes, such as throwing a dart and missing the bullseye. Christians are baptised into Christ's death, and dead bodies don't sin, so sins are not an issue for Christians. There is no more forgiveness. Either you believe in Jesus and are saved or you don't believe and are lost.
Read a chapter of Proverbs every day. Proverbs has 31 chapters so you can keep your place by just looking at a calendar. There is no religion or nothing in Proverbs and you don't have to believe anything. Just read every day. When you are comfortable with that, then read the bible from Romans to 2 Thessalonians over and over until you start to remember what it says. That is the part that applies to Christians. You will notice that most of the religious crap you hear from random preachers isn't in there at all. It is made up.
By disobeying God, Adam and Eve sinned. Being their descendants, we inherited their sinful condition. As a result, we are subject to sickness, old age, and death. Through no choice of our own we have inherited an imperfect body along with sin.
Imagine that you are lying in a hospital bed. You are in a ward where everyone suffers from the same illness, a deadly disease with no cure. When you learn that a doctor is spearheading an effort to find a cure, you are hopeful. You eagerly await news of any developments. One day, you learn that a cure has been found. The doctor behind the breakthrough made enormous sacrifices to make the cure possible.
How would you feel? I'm sure you would respect and appreciate the man that opened up the way for you and many others to be delivered from death. This might seem dramatic, but it really corresponds to the reality we face. We needed someone to deliver us from sin and death. God went to great lengths to do that. In order to save humans from sin, God had to provide a means of atonement, to undoing the effects of sin. So he gave his Son as a ransom for our deliverance. It is an expression of God's perfect will and his great love for humans.
God's original purpose did not change. His will is still for humans to live on a paradise earth, as perfect humans. Christ's sacrifice covers us now if we are willing to accept it and work in harmony with it. Yes, we have free will to make wise choices. Under Christ's sacrifice if we make a mistake or even a serious sin we can repent, and put forth all effort to not repeat that sin again. We don't want to practice sin just because we are imperfect. We want to show God and Christ Jesus how much we appreciate what they have done for us. If you were a father, imagine giving your son to die so that other people might live.
My favorite scripture is found at Revelation 21:3-5, there God makes an unbreakable promise with a guarantee. I find comfort, hope and so much love in that scripture. No, it's not our fault that we are sinner, but we can do all we can for now to wait for the full affects of the cure to take place. No need to make our condition worse by not doing all we can to use our free will to make wise choices to stay spiritually, physically, and emotionally strong, until God brings about the final cure. He's going to make all things new.