Correct. You will enter the information on your W-2 into TurboTax just as it appears on your W-2, or you will likely get efile diagnostic errors that will create other issues. However, you will need to make an adjustment on your NJ return.
New Jersey programming requires that all of the state amounts be added together. However, they should not be added together, because the income is being double-reported to each state. You are able to correct this, however. You will see two screens. The first will ask about your W-2 state wages, and the second will allow you to exclude the NY amounts from your NJ return. I am providing screenshots at the end of this answer to assist you.
In addition, NJ does tax some income that both the Federal and New York returns do not. Please see this NJ website with more information: NJ Division of Taxation - NJ Income Tax Wages - State of New Jersey
This being the case, you will prepare the NJ return first. Since NY also taxes this income, you will get a credit for the tax you paid to NJ on that same income.
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