In NY, can a non-spouse family member adopt a child with the consent of the other parent (consenting to relinquishing his/her rights)? What happens to support obligations in that case? If the remaining parent and/or new, adoptive relative-parent end up on welfare in the future, can the courts come back to the relinquishing parent for money?

Both biological parents would have to voluntairly relinquish their parental rights in order for the child to be adopted. The judge would then have to decide is it in the best interest of the child to be adopted by the family member. If going on welfare is already a concern, that may possibly weigh into the decision. Why place a child with someone that can't financially support them?

If you are talking about one parent relinquishing rights to the other parent, the absent parent would still be responsible for child support.

Once an adoption is granted, all rights and responsibilities for the care of the child are vested in the adoptive parents. The bio parents are off the hook.

Ok, so the bio mother wants you as the bio father to allow a member of her family to adopt the child. She would still be the mother, but this family member, would be the father? Frankly, I don't think that will fly. If she were ALSO to relinquish her rights so the family member and their spouse could adopt, that would work. But just to replace you as the father? No, I don't think so, especially if you don't consent.

To clarify, the mother is suggesting this, but is not suggesting that she would do so. It's just a way for us to part ways. She seems to be saying that she wants my rights and obligations essentially transferred to another one of her relatives (non-spouse, who supposedly is willing). I'm trying to verify whether this is even doable in NY.

You would have to get with an attorney in your area... but I can almost say with 100% certainty... she and you can't do that.

I am in Ohio. When my attorney and I were getting ready to file to terminate my daugthers bio fathers rights my attorney said the laws had changed and a bio parents could not relinquish, nor could rights be terminated, and left solely to one parent. He said someone would have to adopt my daughter in her bio fathers place. I asked could my mother do this (take over bio fathers rights) and he said no. I would have to relinquish my rights as well saying I was unfit to care for my child, then my mother would be able to adopt.

So no, another family member would not be able to take over your rights. If the bio mother marries, and her husband would want to adopt, then you would be able to sign over your parental rights to her husband and allow him to adopt.

But in your situation, you cannont relinquish your parental rights.