It's more than possible to live carless or car-light in NH, people are just too attached to their cars to notice or are not willing to look around for a lot of their neighbors, especially those on the lower social economic spectrum, live pretty carless or car-light lifestyles mostly not by choice and mostly we make it harder for them to do so because "public transit will bring Massholes" or "I don't need it so why do it" mentalities.

Out of experience I know for a fact that is possible, I lived in the southwest corner and between Brat in VT and Keene are both livable without cars. I also had a carpenter friend in Brattleboro that between the bus and ebike with trailer would go to most of his jobs on both side of the border.

Portsmouth, Exeter, Newmarket, and the Tri City (Rochester, Somersworth, and Dover) area on the Seacoast, where i live now, has enough public transit and services in their town center that you can make it without a car as well. They will also have the greenway finished up in the next few years and at least Portsmouth is going to be paving and plowing its section of it with some commitment from Hampton it would be possible to bike from Hampton center to urban Portsmouth on the trail alone.

Lebanon-Hanover-White River is another area where its more comfortable to do so.

Then you have your big obvious ones, Manchester, Concord, and Nashua.