Sam, Eve sinned but man didn't fall until Adam sinned. Go read it... things changed when he sinned because God gave that command directly to Adam. Adam then, as head of his house, gave that command to his wife. She really went against his word, whereas Adam went directly against God's word.
So I don't think her egg or the egg of anyone else is tainted... I think we are dealing with DNA, the code each one of us has that says who and what we will be. And in that code, I believe, is a gene that causes us to die at a certain point.
In the new covenant, it says that God will write His law on our minds and hearts? I think that MIGHT BE idiomatic... I think His law is actually going to be written in our DNA and simply become part of who and what we will be.