I used to fly out of Newark with some regularity with handguns, but it's been at least 10 years. I'd get there plenty early, and bring with me copies of the airline policy on checking guns, and the Federal law that backed up the airline policy...because occasionally you'd get a ticket agent who was convinced that it couldn't possibly be legal, and you need to be able to convince them that they need to check with a supervisor instead of just telling you 'no'. But usually it was straightforward, at least back then.
Assuming they still use the orange 'Firearm' tags that you sign certifying the gun is unloaded, etc, those tags go INSIDE the suitcase. Many years ago they had to go on the outside, which was like a big fluorescent 'Steal Me' sign on your suitcase. That policy changed to inside-the bag quite a while ago, but you don't want the one agent who didn't get the memo to put it on the outside of your bag.
I wonder if shipping the air pistol is an option? NJ would probably consider it illegal, as they pretend it's a real gun, but wherever you are going almost certainly knows better...and the Feds don't care. But that might leave you with the problem of getting it back into NJ, if that is your intent. If it's a one-way trip for the air pistol and if you aren't too far from PA, I'd consider driving across the border and mailing the gun, especially if you can use a non-NJ address for the return address.
It IS legal to ship long guns to yourself...not sure about handguns. USPS and UPS both allowed it, though again the counter clerk will be convinced you can't do it, and someone is going to have to look up the law for them. Shipping to yourself, if you are physically present from where it is sent, and then are physically present to receive, is legal because it is NOT a transfer.