The IRS doesn't care which state taxes were withheld for. The IRS only handles your federal tax return. For state taxes you have to deal with the New York State Department of Taxation and Finance and the New Jersey Division of Taxation.
There is no need to correct your past pay stubs. But your employer can and should make the correction before they issue your W-2, so that the W-2 will be correct. They will issue the W-2 in January. If your description of what happened is accurate, it was the employer's mistake and they should correct it. If your W-2 correctly shows all your income as New Jersey income with New Jersey tax withheld, and it does not show any New York income or New York withholding, you will not have a problem. You will file only a New Jersey resident tax return.
If your employer refuses to make the correction, you can file a New York nonresident tax return and allocate zero income to New York. But since your W-2 will show New York income, you will probably get a letter from the NY Department of Taxation questioning why your tax return shows no New York income. You will have to explain to them that you never lived or worked in New York, and that your employer erroneously withheld New York tax and showed New York income on your W-2. If you can get a letter from your employer confirming the error, it would certainly help.
(By the way, New York has an unusual rule that the state wages in box 16 of the W-2 must be equal to the federal wages in box 1. Any adjustment to the amount of New York income is supposed to be made on the New York tax return. So don't be upset when you see your entire year's wages in box 16 on the NY line.)
If you do have to file a New York tax return, you should prepare the New York nonresident return first, then the New Jersey resident return. However, in your specific, unusual case, it doesn't really matter. The reason for doing them in that order is so that the New Jersey return will correctly calculate the credit for tax paid to New York. But you will have no tax paid to New York, since you will get back all the New York tax that was withheld, so you will not get the credit on your New Jersey tax return.
Whatever happens, make sure that your employer stops withholding New York tax until you actually move to a New York location.
