Since Michigan and Ohio are reciprocal states, you must file a Michigan return and override the income reported to Michigan to $0 to receive a full refund of Michigan state taxes withheld. Ohio will tax all of your income, so you will use your Michigan refund to pay your Ohio tax due.
You also want to advise your employer of this so they can file a form with the state of Michigan that exempts them from withholding Michigan tax but instead withholds Ohio tax.
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