Not the same situation you're dealing with, but I'll get there at the end :)

When I moved to Maine I was still using my out-of-state EZ-Pass (with out-of-state plates) for my daily commute to Boston so I was hitting the York and Hampton tolls regularly. For whatever reason, both states were billing me as if I was paying in cash instead of EZ-Pass, and because I had the auto-refill enabled I didn't catch it for a few months. Once I did, I emailed Maine and New Hampshire to explain what was happening. Both requested some more details, and Maine ended up giving me a significant refund. New Hampshire rejected my request for a refund and didn't provide any assurances it wouldn't happen again. I ended up getting a Maine EZ-Pass and it hasn't happened since. But to me if felt like a bit of a scam: how many tourists going through those tolls are going to complain -- or even notice -- if they get charged $1.25 or whatever more than they should once a vacation. I imagine not many.

All this to say, if you email them with an explanation, this seems like it should be easily resolved. But New Hampshire sucks. :)