Generally, you are taxed in the state where you live and the state(s) where you work.
If the same income is taxed in both places, then the tax is calculated in the nonresident state then brought back to the resident state as a credit. But there is no state income tax in Washington, so you worry only about Oregon.
You file as a nonresident in Oregon. When entering your Oregon data, you should be asked for the amount of Oregon-sourced income. In this case, you enter only income earned for those days that you actually worked in Oregon.
Go through the Oregon interview and see if this works out for you.
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