It seems contradictory, let me try to explain it. God did not create Adam with a sin nature, but because of Adam's sin we were born with a sin-nature. Because of this, the Bible says we all have sinned and fall short of the glory of God. We still have the ability to choose not to, but as God is perfectly holy, the Bible says if we break one law we are guilty of all. God has a zero tolerance for sin. So, even though we may make many right choices in our life, it is inevetible that we will sin along the way. To answer the question is it possible to never sin and get to Heaven without Jesus the answer is no. Only Jesus was able to live a sinless life because he was born of the seed of the woman and the Holy Ghost and did not inherit Adam's sin-nature, and he is God. In Genesis it says the seed of the woman will crush the head of the serpent, meaning that Jesus will crush Satan and evil and sin and death. This was accomplished at the cross when he redeemed (paid for) us from the wages of sin. God does not blame us for something we have no control over, he tells us in the Bible that we have all sinned, that there is none righteous, no, not one. We are simply in a bit of a pickle. But, God loves us and knows because of Adam's and our sin we are separated from him, so he provided the way for us to be reconciled back to him. Since the wages of sin is death, Christ died for us, taking our punishment in our place. This is the greatest truth one can ever grasp, the substitutionary death of Christ. A preacher imagined the conversation on the Mount of Transfiguration where Moses-representing the Law, and Elijah-representing the prophets, were with Jesus. He imagines they told him that all of Heaven is focused on one thing, that Jesus carry out what he came to do to redeem mankind. They wanted to be sure he went to the cross for you and me. And he did.
Even though we have this sin-nature, we do choose to do what is right alot. But one sin is all it takes to separate us from God. His attribute of holiness requires that nothing sinful or 'unclean' can abide with him. The fact is that we were born in a fallen state, spiritually dead to God. God told Adam the day he ate of the fruit of the tree of the knowledge of good and evil he would surely die. He said the soul that sins will surely die. We also inherited this spiritual deadness. The Bible says we must be 'quickened' that is 'made alive' or reborn, regenerated, i.e. born again. That is, our spirit which is dead must be made alive by our having trusted in Christ. It is simply the spiritual condition we were all born into. That is why Jesus said that those who do not believe in him are 'condemned already'. They say, "born once, die twice, born twice, die once." If one is born into this world and is not born again spiritualy, they not only die physically, but then must pay for their own sins having rejected the gift that Jesus paid for them, and go to Hell, ultimately the Lake of Fire, which is called the 2nd Death. If one is born twice, born again spiritually by having trusted Christ, they die only once, physically (unless raptured), their sins having been paid for by the death of Christ and having his righteousness in place of their own, and having been reborn spiritually, possessors of eternal life, passed from death to life the moment they trusted him.
Finally, men ultimately do have a choice. The actual choice is to recognize the fact of their sinful state thus their separation from God and turn to him admitting they are sinners, asking for forgiveness and turning from their way they may think they might save themselves and accept God's way, his provision for the wages of our sin which is death--the death of his son. We owe a debt we cannot pay, but he paid a debt he did not owe, for God SO loved the world. OR, man may choose to reject God's provision for salvation, to choose to let sin reign in his life, and die and pay for his sins himself in Hell. This is the choice. Some may say they will choose not to sin and get to Heaven on their own, but no man is capable. All our 'works of righteousness' are as filthy rags as far as salvation is concerned. We are fallen man and God is holy God, that is just a fact. God provided the way to take away our sins, to blot them out and not 'impute' them unto us anymore by laying them all on Jesus at the cross. This is the way he was able to provide, at great cost. This is his free offer, we may take it or leave it.
