You must file a part-year resident return in each of the two states. Your income prior to your move is entirely taxable by MN; your income after becoming an OR resident is entirely taxable by OR.
You became an OR resident on the day you began living in your new permanent home in OR.
There is no "other state credit" in your situation, because MN does not tax non-resident remote workers.
If they are not already doing so, you should ask your MN employer to withhold OR taxes (and to cease withholding MN taxes). If your employer is unable to do so, you should start making quarterly estimated tax payments to OR, to avoid any year-end underpayment penalties or interest.
This explains how to file part-year resident returns in TT: How do I file a part-year state return? (intuit.com)
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
