I lived in MN for a few months in 2023, but started and ended the year in OH. How should I account for that?

MN says you are not resident if you spent less than 183 days in the state.  Here is the MN reference. 

Therefore, In the "My info" section edit your information.  Indicate your resident state on 31 Dec was OH.  Answer "No" you did not live in another state.  Scroll down to the "Other State Income" topic and answer "Yes" you earned income in another state and select MN.  

This will generate a resident OH and non-resident MN returns.  

Complete the MN non-resident return first and allocate to MN only the income that was earned while in MN.  When complete take note of the income taxed and the income tax applied by MN.  Those numbers will be used to figure a credit for taxes paid to another state on your OH resident return.  TurboTax will automatically make those calculations and you should verify/edit as needed on your OH return.