Re long-guns vs handguns (NY law defines as "firearms" as handguns -- pistols & revolvers), you are correct that licencing is not required for shotguns and rifles. But BTW, rifles, although not requiring a license, I believe, must be registered (different from and less than licensing) in some counties. I discussed handguns and the licensing requirement for them partly because initially I missed the point that the concern was pinking and home defense with rifles and shotguns (I have re-read that), and partly because some discussion did seem to include handguns.
That said, I am concerned that your last comments referencing "prohibited persons" suggests some misunderstanding of NY law. Prohibited persons (special baggage) usually amounts to an enhancement, raising, for example, misdemeanor possession of an unloaded gun to a felony if the possessor has been previously convicted of a crime, or possesses 3 or more unloaded handguns (just 1 is misdemeanor), or possession of an otherwise lawful rifle or shotgun by a person who has been "convicted of a felony or serious offense."
Simply put -- putting aside prohibited persons, or special places -- under NY law a person with a clean record but without a NY handgun license (or other authority, eg, police etc) in possession of (1) an unloaded handgun any place is guilty of a misdemeanor, or in possession of (2) a loaded handgun (or unloaded handgun gun while also possessing ammo) anyplace is guilty of a felony (except in his own home or place of business it's a misdemeanor unless previously convicted of a crime). I didn't mean to belabor a point; the concept that "prohibited person" is required for criminal possession of a handgun in New York State, though reasonable, is not correct. I hope I have not misunderstood what you wrote, and apologize if I have.
Q: "What can I bring to New York State?
A: Not a revolver or pistol.