You should not be claiming the Out of State Tax Credit, in your situation (part year resident). You just need to allocate your income as MO or non MO income.
Try this: at the W-2 screen, in the federal section, for your W-2 from working in WA, put WA in box 15 and the box 1 amount in box 16. This should help TurboTax allocate you WA wages as non MO income.
