Go to court first thing tomorrow morning and file in Ex Parte motion asking the Court to find your child support is current and to cancel the capias. Attach proof of payment to your motion. The clerk will be able to help you with the procedure of filing. In your motion you should explain the urgency of the matter and the irreparable harm that could result if the relief you requested is not granted. The judge may wish to speak with you but it's more likely that you will get a ruling on the motion without a hearing.
I'm not admitted in NH, but your question caught my eye as I've done a work on this issue here in NJ.
Actually, if I were you, my first call might be to a civil rights attorney. If they arrest you erroneously, you've probably got a lawsuit against the state - it depends on your state's law and various immunity provisions.
That said, the advice about going to the courthouse physically with a copy of the notice and proof of payment is probably the best way to quash it.
IF YOU LIKE THIS ANSWER AND APPRECIATE THE TIME IT TOOK TO WRITE IT, PLEASE SELECT IT AS "BEST ANSWER." Thanks. The above is said without seeing your case file and without my understanding the entirety of the facts of your case. Depending on those facts, the above information be may incomplete or may be completely inaccurate. The above is intended as general information only based on what you described and not as legal advice. I advise you to consult with counsel who may be able to provide better information commensurate with a better understanding of your situation.