I was born and raised on Long Island and for 39 years lived in or near NYC, Philly, Baltimore, and Washington DC.
I don't feel isolated in NH -- I feel like I have finally found space and peace.
It all depends on your outlook and what you are looking for.
That said, OP, if you want to feel like you are in a great little town/city, do look in Keene proper. Brattleboro has a similar vibe but you have the winter commute to consider..not a deal-breaker for all but would be for some.
Also, politically and tax-wise, VT and NH are very different. VT is way more liberal/crunchy and NH is more libertarian. No income tax in NH combined with no property tax might be really appealing to you...
But -- especially compared to NY politics, you might find that neither state is too extreme compared to what you are used to. And things that can sometimes raise eyebrows in NH [dreadlocks, body piercings] that you see in Keene and Brattleboro and Burlington barely even register on my radar -- I don't even "see" it.
For the most part it's a pretty accepting, welcoming place as long as you don't try and turn it into the place from whence you came.
Welcome to NH - I hope you love it as much as I do.