You may only have to file for Ohio personal income tax return. As you said, Tennessee does not have an income tax. The state only taxes interest and dividends. The amounts have to be over $1,250 ($2,500 if married, filing jointly) during the tax year.
File a part-year Ohio tax return and report only the income you earned while you lived in the Buckeye State.
If TurboTax prompts you to file a TN return and you don't need to, select Skip.
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