If you have LEOSA, you should be exempt from pretty much every specific law, but really they'll only do that outside the NYC area (most SPs and sheriffs would give you leeway outside city limits). There are a number of stories out there of even active duty cops traveling to the state or city and back (e.g. for prisoner transfers, etc), TOLD them they were coming, showed local cops their shield and still ended up getting hassled for carrying. If you have signed documentation of this stuff, definitely keep it in the glove box if and when you end up getting stopped for something.
But yes, even if you qualify for LEOSA or your destination is through NY to another state where you can legally have / carry whatever you have with you (peaceable journey) NY is still well-known for stopping folks and still forcing you into court to affirmatively defend via federal statute, even if you put guns and ammo in separate locked containers in non-accessible areas of your vehicle. As long as you travel through NYS to somewhere else where you can legally possess / carry (and you don't stop for more than food/gas on the way through the state) then you're still legally ok, but some jurisdictions may choose to make you go to court anyway.
All that said, if you don't have a NYS permit, and you do the above things and avoid NYC, you should be fine. However, if your destination is IN the state and you have pistols (and no NYS or NYC permit), you do run the risk of a class E felony.