Manalapan is usually pretty good. Its been a while since I applied for permits but 30 days or so was typical. Expect a little longer now.
Just watch email for notices or anything you need to do. Contact Lt. Drucker if you are not hearing back. He is usually in 1 day per week. He is retired but covers Firearms. Good guy.
A good local shop is Howell Gun works. Inventory is low everywhere, but if you want “something”... like a simple 12 ga pump shotgun, they will probably have something like that on hand.
Also check out the FFLs listed here in the forums. Many good FFLs that operate small shops or out of homes (yes...legally) and they may have what your looking for. Just ask here.
If you use an out of state shop like Heritage Guild in Easton, PA, they will have to transfer the gun to a NJ FFL that you pick. I find it easier to just shop in NJ.
Avoid Dicks Sports. That company is an enemy of the firearms community. Just my opinion.
If you need a place to shoot, there are a number of Public ranges. But you can also join clubs like Old Bridge RPC and Central Jersey RPC. Both clubs also have competition shoots open to the public that help newer shooters, but get some practice and or training first if you are new to shooting. Its worth the time.
good luck and welcome
-Kevin