The issue is really about can you reasonably expect an LEO in another state to know the laws of your state.
In most cases, this is a NO, you cannot reasonably expect an LEO in another state to know the laws of your state.
So, while you do not "legally" have to show proof of insurance or financial responsibility prior to an accident in New Hampshire, and that "follows you" to other states, an LEO in another state is not going to know that (and WILL NOT "just take your word for it") and will ticket you and let you deal with it in court.
I would suggest that if you plan on driving in other states a lot, that you talk to an attorney about the benefits/risks of carrying something like a notarized statement of financial responsibility to show in other jurisdictions, etc.
Otherwise, you will face this again and you may not have the luxury of a zoom hearing next time. And you can be pissed about it all you want, but it's perfectly legal for an officer to ticket you for a violation in that state and let you argue it in court.
P.S. It sounds like your vehicle is registered in Florida while you are licensed in New Hampshire. If this is true, then this is likely creating a greater chance of your being ticketed - most LEOs are going to question someone who is licensed in one state and driving a vehicle registered to them in another state - it raises red flags for them.
