Your understanding is correct.
Since you earned income from Kentucky sources, you do need to pay taxes on the income earned in Kentucky when you were a resident and nonresident.
Boxes 15-17 of your W-2 forms should reflect your state wages and withholdings. If you did not have enough and are still working there, you should consider reaching out your employer to change your withholding amount to avoid being in the same situation next year.
Also, you cannot claim a credit in Washington as you said. Washington does not have state income taxes. The credit is only there to assure you are not taxed twice on the same income.
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