After you file

We can't see how you filed, but if you filed it wrong, you can correct it by filing an amended return.

Since Washington doesn't have a state income tax, you would not pay income tax on any of the income earned in Washington while you lived there. If you only had W2 income in Washington, there is probably not a need to file a return in WA, but since I don't have all the information on your situation I can't confirm that for you. For Oregon, you would need to file a part year resident return and pay tax to Oregon on the income you earned in Oregon. So the first thing verify is that you filed a Part Year Resident Oregon return, Form OR-40-P 

TurboTax asked you how much of your total income reported on your Federal return was earned in Oregon or while you were a resident of Oregon. Hopefully there you entered the amount of income you earned while living and working in Oregon. If you look at page 3 of your Oregon Return you will see two columns. On the left is Federal and the right side is Oregon. Ensure only your Oregon income is in that right column. Since you don't say, I'm making the assumption you only have W2 wages, so that is the only thing I am addressing.

If you filed a Part Year Resident Oregon return and only the Oregon wages are in the right hand column on page 3 of your return your return was likely filed accurately. If not, you will want to file an Amended Oregon State Return.

Here's how: How do I amend my state tax return? - TurboTax Support - Intuit