I must admit that I don't much about CT handgun laws. Parts of CT are very close to NYC, so it's worth looking into for you. Some NY info:
--Any handgun permit, even a premises-only permit, is going to be virtually impossible to get in the five metro boroughs of NYC. And in NYC, even your long guns will have to be registered.
--People who work in NYC but don't live there have to pay NYC income tax, but they don't have to pay NYS income tax if they live in another state.
--Outside NYC, permits are issued by the individual counties, but are valid anywhere in the state (outside of NYC).
--Westchester County (you mentioned Sleepy Hollow) is not too friendly to CCW. All handgun permits are automatically issued as restricted (target and hunting only), and it's pret'near impossible to get the restrictions lifted. You have to prove "need" for a carry permit, and they'll deny you on any whim they can think of. Rockland County, on the other side of the Hudson River, is just as bad -- ironic, since it's where the Kahr Arms main office is located.
--You have to go at least one more county north to get into more gun-friendly territory: Orange County on the west side of the river and Putnam on the east. I'm in Ulster County, which is above Orange--about 1.5 hours from Manhattan. Up until recently, it was relatively easy to get an unrestricted permit (full carry) in Ulster County. All you had to do was include "self-defense" on your application where it asks why you want a permit. In the past few months, though, due to a change of judge, all new permits have been issued as restricted. Whether they can later be amended to unrestricted remains to be seen.
--Long and short is that you'll probably have to be more than 45 minutes from NYC if you want to live in NY and have a carry permit.
EDIT TO ADD: What instrument do you play? If you end up having to justify your need for a carry permit, could you say you routinely travel with an expensive instrument? Carrying large amounts of cash or jewelry is a valid reason. Why not your Stradivarius?
EDIT TO ADD: NJ residents, please chime in, but I'm pretty sure you can forget about getting a carry permit in NJ too.