"I live in NJ but work in NY." - TurboTax will handle this - you won't be double-taxed.
Generally, you owe tax to the state where you live and to the state(s) where you work.
In this case, you will owe tax to NYS because you work there.
You will enter data for NYS first as the nonresident state. Then you will enter data for NJ, your home state.
On the NYS return, tax will be calculated and your withholding for NYS will be netted against it, so you may or may not owe NYS.
But when you do your NJ return, even if it appears that NJ is taxing the same income, there is on line 43 of the NJ-1040 "Credit For Income Taxes Paid to Other Jurisdictions", i.e., the amount you paid NYS is used as a credit in NJ.
This is how many states avoid double taxation, by giving you a credit in one state for taxes you paid in another state on the same income.
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