Welcome! We moved here from NJ about 5 years ago and waited about a year or so before sending in our applications.. wife and I. In the meantime we got to know our neighbors, learn the area, where everything was and pretty much getting settled. When I started to ask for references I was honestly surprised and how open people were to it even though none of them carried themselves. We actually wound up with a dozen or so people willing to sign for us.
Make sure you order a complete detailed NJ DMV Drivers Extract and include it with your application. No, its not listed as a state requirement however the Federal Investigator running our background checks contacted us and said they required it for all new residents. IIRC it was all new DE residents of 5 years of less they are requesting it from. There are two, a basic for employers and then a more detailed one. The Federal Investigator running our background checks required the detailed extract. It added an additional few weeks to our wait.
It's a good idea to take your CCDW course beforehand and submit the certificate with the application to speed things up. Lots of people here have taken the First State Firearms Training class in Bear. Its a great 1 day, decently priced class instructed by David and Jeff (father and son). Wife and I took their class and found it fun, informative and well taught. She had only fired a handgun once before and they were great with both in class and at the range. They used the state run range Ommelanden for the course and then the class portion at their home.