You are on the right track. You have to use a technique called allocating. When you allocate, you designate the amount of income that you earned in each state. That way, you pay the correct amount of tax to each.
When you file the part-year returns for KY and OH, you will run into a screen that asks how much of the income was earned as a KY resident and then as an OH resident (on each return). The amount reported on your W2 will pre-populate, and then you should be able to either override the amount or simply input into the corresponding box.
What you are seeing on the KY return is correct because you should receive a refund, since not all of the income was taxable in KY (only the portion of the year when you were a KY resident). You should owe money to Ohio (since no OH state tax was withheld), but your KY refund might be higher than what you owe OH.
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