You don't need to correct your W-2 if your wages belongs to New York. You still file a New Jersey tax return as a resident.
When you file your New York return, your total income will show up in New York as taxable, but once your tax is calculated, it can be reduced by a percentage of your income earned while out of state.
In New Jersey, you will get credit for any tax paid to New York. So, you won't get double taxed even though it may appear that way at first.
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