If you are asking for LEGAL advice, it is not a simple matter to "give up your rights" to your child. In fact, most States don't allow it unless the other parent is in a stable marriage and that partner is willing/able to adopt the child.
If you ARE able to "give up your rights" you are still responsible for child support. The child still needs to eat, wear clothes, etc.
Of course, you don't actually have to exercise your rights to visit so perhaps that is a solution.
The personal/relationship advice - I'll leave that to someone else.
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