Your annual reward would be taxable if it was earned due to your New Jersey employment. That’s why your employer withheld NJ tax. If you have questions you can speak with your HR department. It’s the same as if you quit your job and moved and your employer paid out unused vacation.
I suggest reporting whatever is listed as NJ wages in Box 16 of your W-2. However, if you think only half should be NJ income, the non-resident return allows you to state how much income you earned in NJ. If the amount is different from your W-2, you may get a letter from NJ, but as long as you have some type of documentation, you won't get into any trouble. You'll just have to respond and explain the difference.
Normally you get a tax credit on your home state return on double-taxed income, but since Washington has no income tax, you cannot claim a credit for tax paid to NJ or CA.
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