There should be a caseworker assigned to your case. Put the facts as you just wrote them in a letter to the caseworker. You also have a right to request an audit of your payments - this should result in a showing of your overpayments. I imagine the child support office will credit you with this amount rather than issue you a check. Do not hesitate contacting your caseworker often until you get a response.
The only ways you can get the money back is to talk with the mother and ask her to give back the excess money that was paid (good luck with that!), or file a motion with the court asking it to order her to return the excess money. You will have to have a hearing and show the judge how much you have overpaid. Or, if you are still paying child support (it isn't being withheld from your paycheck), you could just make several smaller payments until you were even.
This information is provided strictly for information and is not intended to establish an attorney-client relationship between the attorney and the party asking the question.
You can request a formal accounting to verify your claim. If a phone call does not work, you can file paperwork in the R case (child support case) to request this.
I agree with my colleague that you will likely not receive the money back, but that you will be credited for future payments. If you have no future payments, then they will have to refund the excess.