New York does NOT recognize the Federal Transport Exemption. As a result if you enter the state of New York with a handgun in your possession you are immediately guilty of the possession of an Unregistered Handgun, a 5 year Felony.
New York also does NOT recognize any other state carry permit, so in New York your carry permit is nothing more than a potential alert to the New York State Police that you may be in possession of a Handgun. BTW, MA. CT, & NJ, also do NOT recognize your permit but usually do recognized the Federal Transport Exemption.
I would advise that you forget about taking a Handgun on your trip. Because there isn't any means of getting to New Hampshire, Vermont, or Maine over public highways that doesn't require transiting through New York.