Call the employer and tell them that you received your tax forms, but they are incorrect and request that they reissue it correctly to reflect that all of your income was earned in New York because that is the only place that you ever worked.
Give them 30 days to reissue the correct statement. If they don't, then call the State Tax Department of New Jersey and New York and tell them what happened: That the company you worked for in New York, moved to New Jersey and that you only worked for them in New York, and that they issued you a tax receipt for the State of New Jersey. Provide each of the State Offices with copies of each of the tax receipts.
I remember some years ago I used to telephone solicit for an insulation company. My hours were fixed and I worked out of their office, using their telephone. Yet when it came time to give me a tax receipt, they refused. They claimed I was a private contractor. When I insisted I wasn't and demanded a tax statement, they fired me and refused to give me my final pay statement of earnings I had earned the prior 2 weeks.
I contacted the tax office, explained the situation. They asked if my hours were set by me (they weren't), if I worked from home if I wanted to (I couldn't and didn't). According to them I did not meet the requirements of being a "private contractor", and that I should be getting a tax receipt.
The whole situation turned into a whole big mess. It was so long ago I can't recall all of the details, but I remember that I ended up taking them to small claims court, and won. He appealed it and lost, and reappealed it again. And some time later, when I called the courts to make an inquiry about something I needed for my part to prepare for the reappeal, they had conveniently "lost" the entire case/claim. As a result I never got the money they owed to me from my prior 2 weeks work. The interesting thing is that one of his in-laws worked for the courts. And I'm sure had something to do with everything having gone missing.
The guy ended up closing up his business because while you can muck with the court system if you have someone on the inside, you sure can't do anything when you have the Provincial and Federal governments breathing down your neck. So he got his just desserts, even though it cost me 2 weeks wages.
So far as my own tax return went, the government helped me with my tax return that year and averaged out the deductions etc.