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You do go straight to court in NH for traffic violations. The officer is supposed to show up to testify, but if he cannot they will potentially send a prosecutor in place. If your experience is anything like mine (at the Keene District Court seeing as how the violation was in Cheshire county), the prosecutor will probably come over to you before you stand before the judge and give you an offer. In my last case several years ago, the prosecutor asked me to sign a piece of paper that said they agreed to drop all charges if I did not get any more violations within a years time, and if I did get anymore violations they would tack on the original charge to it.
Once I signed that, I stood in front of the judge and told him we made the agreement, and I was off on my merry way.
Or you might get lucky and have nobody show up at all, at which point you can ask to question the officer...which would probably get you off.