The New Jersey part-year and nonresident filing threshold is based on gross income from everywhere for the entire year, not just NJ income.
See Part-Year Residents.
You would have to file a NJ return if your total income was more than $10,000 if filing single or $20,000 if filing as a married or civil union joint couple, head of household, or qualifying widow(er)/surviving CU partner.
You are probably not seeing NJ because TurboTax only gives you one free state. Tap Purchase State. Since you did not start California, you may be able to use NJ as your free state.
If you lived in only NJ and Washington in 2022, I don’t know why you have California, unless you also worked in California.
I don’t know where TurboTax is asking about other tax situations.
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