You will have to file both KY and OH returns. The state tax interviews will allow you to allocate your income between the two states based on the amount of time you were a resident of each state. You should get a refund of OH taxes on the money earned after you moved to KY, and you will probably have to pay KY tax on the money earned after you moved to KY .
Ohio and Kentucky have reciprocal agreements. Your employer should be able to withhold KY taxes from your pay and not withhold OH taxes. You would need to contact the payroll department and submit IT-4NR.
