Welfare public assistance and child support enforcement issues are highly technical and confusing even for the attorneys' who work in the field. If you are currently low-income you may qualify for assistance through your local federal Legal Services Corporation funded agency. I recommend that you contact that organization for an appointment as soon as possible. In terms of the basics of your case as described you should understand the following:
a. The date that the county/state is seeking support for your children is based on the date that the children began receiving assistance. Also, the fact that you yourself were low-income and receiving welfare assistance does not wipe out your child support obligation. This most likely became a month that you were not able to pay support and for which an arrearage and interest were calculated. Generally medical care costs is also calculated through a separate medical support order. One more thing, your former wife was required to assign her rights to child support as a condition of receiving assistance.
If your son now has private medical coverage, what you are now being held liable for are arrearages (past due) child support including medical support for any month that the child received any welfare cash assistance. Your son can probably not be taken off of his mother's welfare assistance grant if he has any half or full sibings who are also receiving assistance under a welfare rule known as "sibling deeming". This means that any money that is available to support one child, with some exceptions, like SSI, is considered available to other children in the same household.
One final thing: this is not your wife's doing but rather a matter of federal welfare law as administered by the state of New York. You and your wife may need to have separate attorneys especially if you are being threatened with welfare fraud. You also do not want to discuss this matter, given that threat, with anyone who is not an attorney who has been hired or retained to represent you.
Sorry that this is so complicated but welfare rules are not simple and the rules are hard to explain without knowing very specific facts.
This is not a substitute for a consultation with a family law attorney who has a background in the connection between child support and welfare rules.