As a full years resident, you pay MI income tax on all your income, regardless of where earned*. If some other state also taxes some of that income, MI will give you credit, or partial credit for the tax paid to the other state.
It's not clear what "I paid Ohio state tax for a house I still own in Ohio" means. Is it rental property?
*There is an exception for wages earned in reciprocal state (OH is reciprocal with MI). Separation pay taxed by OH, for work performed, prior to you becoming a MI resident would not be covered under the reciprocal rule. You would have to file a non resident OH return, then claim a credit on the net OH tax paid (not the withholding on the W-2). You would include the rental income (if any) on the OH (and MI) return.
