I've seen folks mention using https://www.shipmygun.com/go/ with good results, but they're not from NJ. That said, I think you can request a quote so you can see whether you can use it here..
I just recently shipped a rifle to a buyer but I did it via my FFL. I brought it to them w/ the box the rifle came in and they packed it up in another box and shipped it for me. Very straight forward.