If you are truly traveling THROUGH New York State, this is exactly what the FOPA is designed to protect. It might be a good idea to print out and carry with you a couple of copies of the actual FOPA, from a legal web site, not from a 2nd Amendment web site that paraphrases it.
Remember that in order for FOPA to apply you must be legally entitled to possess the gun in the state where the trip begins, and in the state of destination. And "through" apparently means "through." If you stop to eat at Mickey D's along the highway, that's in the normal course of travel. If you get off the highway and drive 50 miles north to have lunch with Uncle Joe while you're in the area ... that's starting to look more like going TO New York rather than just "through" New York.
And, as noted, stay clear of NYC. Wherever you are going from and wherever you are going to, there are routes available that will avoid the City, but they may be a bit out of the direct route.