I googled "what happens if I move while collecting NJ unemployment and found this:
Q. What should I do if I move?
A. If you move while claiming benefits, you may report your new address and new telephone number when you claim benefits on the Internet, by telephone, or by mail. If you worked in New Jersey and have relocated to another State, you must file a claim against New Jersey. You may file your claim by calling the New Jersey claims office that handles out-of-state claims at 1-888-795-6672.
If you move out-of state while you are claiming benefits, you should immediately telephone 1-888-795-6672 and select the option to file a claim in order to report the address change. An agent will assist you in claiming benefits against New Jersey on an interstate basis. If you move to a neighboring locality in New York, Pennsylvania or Delaware (within commuting distance of New Jersey), you may change your address by telephoning a New Jersey Reemployment Call Center at the regular telephone numbers.
Here's the link to the site (NJ Department of Labor) Department of Labor and Workforce Development | Frequently Asked Questions