First you need to make sure that any ongoing child support order is terminated, which you can do through the DA's office (or whoever is doing the collecting now). Then you need to establish a support order against the birth mother. She owes $100 per month per child even if she is still in jail or unemployed. From there, you may be able to offset what the birth mother owes you as against what you owe her/welfare. Finally, you need to make sure you have a court order granting you custody. You cannot risk that birth mom pops back into the scene and claims custody along with child support. See a lawyer to review your existing orders and determine how best to proceed. Feel free to call my office if you would like a consultation.
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You need to go to court and ask for a modification of the child support order based on the change of circumstance. Generally speaking child support arrearages are not eliminated, however if you can show your child(ren) were living with you during a portion of the arrears period, it may help you. I suggest you contacta local family law attorney to discuss your options. Best wishes to you.
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You need to file a motion either in family Court or with the District attorney's office so that they know you have had custody. It is unlikely that they will modify your obligation retroactively, so you must file immediately. There is not enough information in your question to know the economic viability of hiring a lawyer.