TN has no state income tax so you have nothing to file in TN. But you lived and received income in OH for part of 2024, so you will need to prepare a part-year return for OH and allocate the amount of income you earned in OH to seek a refund of excess tax withheld for OH. Going forward, make sure the employer has stopped withholding OH state tax from your wages.
MOVED OR NEED TO FILE PART-YEAR STATE
https://ttlc.intuit.com/questions/1901560-how-do-i-file-if-i-moved-to-a-different-state-last-year
https://ttlc.intuit.com/turbotax-support/en-us/help-article/printers-printing/file-part-year-state-r...
https://ttlc.intuit.com/turbotax-support/en-us/help-article/printers-printing/file-part-year-state-r...
ALLOCATE PART YEAR
https://ttlc.intuit.com/turbotax-support/en-us/help-article/state-taxes/allocate-split-income-part-y...
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