This is general information and not legal advice. No attorney-client relationship is formed. This is for educational purposes only.
Yes, most likely you can terminate your child support obligation, possibly even for your daughter. (In New York, usually child support obligations are to age 21.) You don't have to wait to pay off the arrearages to move for a termination of child support, at least for your son. You can file a motion in the family court where child support was awarded initially setting out the facts you cite in your question: your son is 23 and self-sufficient, your daughter is an adult and not pursuing further schooling and you are now on public assistance.
Although it may not work, you can ask the Court to waive the arrearages. It's uncommon, but it doesn't hurt to ask, and your situation is one the Court will see sympathetically, especially since you have paid off that large amount of arrearages in the past.
Here is a link to Do It Yourself forms on the New York courts' official website: http://www.courts.state.ny.us/courthelp/diy/familyCourt.shtml
The tool you'll want is the last one, "Support Modification Petition Program."
In addition, here is a link to all New York State child support forms: http://www.courts.state.ny.us/forms/familycourt/childsupport.shtml
The federal government recently adopted the Access to Justice initiative to make courts more user friendly. You may find your court's self-help option helpful. You can access it here: http://www.courts.state.ny.us/courthelp
Then under "Families & Children" click on "Support."
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