The law in the state of Georgia is that child support arrears may not be modified by the court or waived by the custodial parent. That being said, there are sometimes ways to do things such as this if the parties consent. The big problem here is that the child's mother has applied for Medicaid and that brings the government into the equation. I don't know why the state would require her to collect arrears from you if she wants Medicaid for a child that is not yours. You might want to get some clarification on that issue.
That being said, I would strongly recommend you speak to an experienced child support attorney in the state of Georgia. As I have said, there are ways to do what you want to do, but you have to get the paperwork right.
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The State of Georgia does not authorize the parties to modify child support on their own. Only a court order can do that. Also, the Judge is not authorized to retroactively modify child support obligations. Your son's mother may choose not to actively pursue the arrearage. However, she cannot forgive it and make it go away. When she seeks government assistance, such as medicade, she is subject to their rules.
You will need an attorney to draft a Consent Order for a Judge to sign regarding the collection of the arrearage.
I am exclusively a family law attorney, practicing primarily in the metro Atlanta, Georgia trial courts. However, I handle appeals from anywhere in Georgia.
You cannot "drop" child support arrears. The only option is for the parent who was owed the support to not pursue collection of it. Unfortunately, however, the custodial parent is generally required to pursue collection when they receive government assistance as the view of the "state" is that the assistance may not have been needed had the noncustodial parent paid their support.
~ Kem Eyo
The above answer is a general explanation of legal rights and procedures. It does not constitute legal advice. Nor does it establish an attorney-client relationship between the individual posting the question and the attorney providing the answer.