Ah . . . . the fantasy of every deadbeat parent. No, he can't "give up his right" to avoid paying child support. And since the right to be supported belongs to the child, letting him get away with not paying would be the wrong thing to do. By and by, when you have a stable marriage to a responsible man who wishes to voluntarily adopt your child, the rights of the bio-father can be terminated. When an adoptive father willingly assumes the support obligation, the bio-father will owe no further support.
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The short answer is yes, assuming that the judge also finds that termination is in the best interest of the child and that there is an adoptive father. Also, even if he gives up his rights, it does not release him from BACK child support.
Termination of parental rights is a very serious action and the statutory grounds are found in the Texas Family Code, Chapter 161. You'll have an easier time meeting that burden if there is a stable man wanting to adopt. Regardless, the court must order a social study. A criminal background check will be conducted on the adoptive father. Besides paying the filing fees (around $300), you'll have to pay for the attorney appointed to try to find the father and you'll have to pay for the social study.
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Unless your husband is adopting at the same time, most courts won't terminate parents rights.
Do what you can to help father and child spend time together and build a relationship.
This answer DOES NOT establish an attorney-client relationship. This answer is based on the limited information provided and is not intended to be conclusive advice. There are likely other factors that might influence or change the advice after a more lengthy consultation.
He would need your consent to give up his rights. The courts won't terminate just because he wants out of child support. But if you do terminate rights then, yes, it would cut off any future obligations for support but not necessarily what he owes in arrears.