First concept to understand is that NJ is going tax all your income from all sources as your resident state. You will claim a tax credit on your NJ return for the income taxed by the NY company.
If your wages in box 1 are 100K you will have a NY state line (boxes 15-17) that reflects the full amount of box 1 wages in box 16. Then you will have one NJ state line (boxes 15-17) that will also have the total box 1 wages in box 16 and shows the total withholding for NJ from all employer forms.
Next complete your NY non-resident return where you will allocate the 50% of wages to NY.
Then complete your NJ resident return and you will arrive at a page where you select the NY wages as duplicate.
After that you will arrive at a page to verify the credit for taxes paid to NY which will auto-populate based on your NY return info.
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