Can a man that is already on child support sign away his parental rights & be done with paying child support?

My fiance' has two outside children and is on child support. He is behind like $10,000. But due to this he has been arrested and his license has been revoked. Well he's been looking for a job but can not get one due to his situation and the debt keeps growing. The mother of the children won't allow him... show more My fiance' has two outside children and is on child support. He is behind like $10,000. But due to this he has been arrested and his license has been revoked. Well he's been looking for a job but can not get one due to his situation and the debt keeps growing. The mother of the children won't allow him to see the children and she asks for stuff and he gives it to her if he has it. When we presented this to the courts they said its all gifts and can't be used towards his payments. Of course the stuff he gives them comes from me because he doesn't have a job. I have stopped giving them stuff now because it makes no sense if she has him on child support and still wants other things. I am post-poning the wedding because as I understand it, if he falls even more behind and we are married then they can start garnishing my checks as well. This is so not fair. He's a great guy with a bad situation. He hasn't seen the children in years and have no bond with them because of her. We have made attempts to get the kids for weekends but she refuses. Honestly he wants to just do away with the whole situation. I know it sounds bad. But legally can he give up his rights and just pay the back child support and stop further child support from accruing?
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