can the father stop paying child support if he moved to another state and does see the kids

Would there be deadbeat parents if they can simply "give up [their] parental rights, and stop paying child support"?

No one can force you to exercise your parental rights. However, the state will force you to meet your parental duties among which is to pay child support. If your child support order originates from NY, my understanding is that in NY child support is ordered until the child is at least 21. You can check this with an attorney in the state that issued the child support order.

If the children's father marries another person and the new wife wants to adopt the children, you can agree to the adoption. Once the adoptions are legal, your duty to pay child support would end.