check out ocshooters.com on their you should find a list of the judges and who take care of which area. also technically there are only two permits to possibly get. a carry permit and a premises permit. im going to guess you are applying for the carry permit. with the carry permit the judges have the power to issue restrictions limiting the time you are allowed to carry to things such as target/ hunting restriction. a restricted permit is more common than an unrestricted permit ( which would be the CCW you refer to) but it also depends on your judge and your reasons for needing the permit. good luck
Technically all carry permits in NY are CCW, some judges will just put restrictions such as hunting, sportsman, etc on it. If you ask for unrestricted the worst that will happen is they will put restrictions on it (assuming it will be approved anyway). Good luck.