NJ law is subjective at best...
Outside residences coming in with a firearm, if you have a specific reason for a long gun or handgun (range or competition), you are fine... judging you have magazines 15 rounds or under and the firearm isn't an Assault Weapon or banned by name (can look up NJ law, but very similar to the old Federal AWB). Other exception being LEOSA, but if you don't fall under that, gun goes straight to the intended location and then back out of state. Until they figure out this "reasonable deviation" BS, you kind of need to consider that driving aimlessly around might be an issue if you get pulled over/firearms are discovered.
That is the same for NJ residents and handguns. You can't go to the range with your pistol, then run errands with it in the trunk... in the way the law is written. The same goes for NJ residents with long guns, without a FID.
If you have a NJ FID, which I believe outside residents can get (pointless unless you are bringing long guns into NJ regularly or will buy long guns from NJ FFLs), you can leave a long gun in your vehicle and do figure-eights around the cloverleaves of any major highway you want. No problems with going anywhere, but do remember... if your vehicle is broken into, you have a firearm on the street. In regards to casing it and securing ammo, that is left to your best judgement for long guns... but if you get into a major accident, having shotguns and rifles skewed across the road isn't the best thing for an officer/Trooper to arrive to. Same to consider if you get pulled over, do you really want to draw attention/deal with the hassle for having an uncased long gun in the back seat? And if the person pulling you over doesn't know the law exactly, probably going to be jammed up for it.
The horror stories you hear tend to be people carrying a handgun legally in another state, crossing the border, and being stopped/admitting they have a firearm on them. While I think it is a stupid set of laws, you should know the laws for any state you carry in. If you are nearby a state that is very aggressive towards firearms, it would be a good idea to look up the laws, and make it a point NOT to bring a loaded firearm across the bridge.
But you also get people that think everything in NJ is illegal. If you own your own business, you can legally carry a handgun at the place of business (has to have a physical, stationary location). You can legally carry while on your property... like cutting your grass. Start waving the gun around at people, then you are breaking the law...