Ohio and Michigan have a "Reciprocal Agreement" where the employee only pays State Tax to the state in which they live.
You should let your employer know which state is your resident state and that is the state tax they should withhold for you.
If you live in Ohio, you only pay Ohio state tax on income earned in Michigan and/or Ohio.
If you live in Michigan, you only pay Michigan state tax on income earned in Michigan and/or Ohio.
If state tax is withheld for the wrong state, a non-resident (or in your case part-year resident), state return needs to be filed to get the tax paid erroneously refunded.
There is a form you can give your employer so they withhold tax for the correct state
NOTE: If you work in a CITY or COUNTY that taxes, that tax needs to be paid. The agreement is only between the States.
Find the form you need.
[Edited 02/11/2022 I 4:28pm PST]
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