Legal Question in Family Law in New York

No. I'm overly simplifying this, but unless there is some other man who wants to be declared the father of the child, or unless there is some way for you to challenge that the child is not yours, then you are out of luck.

No. "Signing over right" -- which is not a legal term - presumably means agreeing not to see the child or participate in decision making. You certainly can do that - but it does not relieve you of the legal obligation to support your child. And, notwithstanding the previous answer, if the child knows you as the father simply challenging paternity does not relieve you of support obligations either.