Yes, as a resident of another state and non-resident for NY you would owe NY state tax on income earned for work carried on in NY. Here is the NY reference.
The address on your tax return is the address where you receive your mail on a regular basis. Typically that will be in your permanent resident state. The address on you tax return is not shared with anyone; it is for tax correspondence purposes not residency determination.
To generate the proper returns in TurboTax in the "MY Info" indicate that you lived in your permanent resident state on 31 Dec and if you did not have another resident state in the tax year indicate "No" you did not live in another state. Still in the "MY Info" section scroll down to "Other State Income" and answer "Yes" you earned income in another state and select NY from the dropdown list. This will generate any resident returns needed as well as a NY non-resident return.
Your resident state, assuming that state has an income tax, will tax your income no matter where it is earned. NY for non-residents, only taxes the NY source income. For any income that is double taxed, your resident state typically offers a credit for taxes paid to another state. In TurboTax complete the non-resident state return before the resident state return and the information needed for the credit (income taxed and tax liability) will automatically transfer as needed.
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