Since you work in NY, you should be having NY tax withheld from your paycheck, and not NJ tax. Your pay for working in NY is fully taxable by NY. You will owe little or no tax to NJ because you will get a credit on your NJ tax return for a large portion of the tax that you pay to NY.
Don't worry about the NJ tax that has already been withheld this year. You will get it back when you file your tax returns at the end of the year. You won't be double taxed. But you should have your employer correct your withholding for the rest of the year, and do it as soon as possible. You should consider having extra NY tax withheld for the rest of this year, to make up for the NY tax that should have been withheld on what you already earned in NY this year. If you do that, make sure you remember to remove the extra withholding at the beginning of next year.
When you file your tax returns at the end of the year you will file a NY nonresident tax return and a NJ resident tax return. Prepare your NY nonresident return first, then your NJ resident return.
