State tax filing

Your understanding is correct.  You only need to pay Missouri taxes on the income earned when you were a resident there.

Missouri allows you to either claim a credit for taxes paid to the other state before becoming a resident by using Form MO-CR, OR you can use Form MO-NRI to just pay taxes based upon your Missouri income percentage.

Since Washington does not impose income taxes, you should file Form MO-NRI. 

You will need to confirm your input in the My Info section of the program to ensure you are preparing the MO return as a part-year resident.

1. Click "Edit" to the right of your name 
2. Confirm your input under #2 "Tell us the state(s) you lived in" 
3. Confirm the rest of the state-specific questions here as well including the "previous state of residence" and "date you became a resident" of the state on December 31, 2019. 
4. Do the same for your spouse if you are married filing jointly. 
5. Go back to the personal information section, and verify the mailing address and "Other state income" section. 

Once you enter this information, you should go back into the W-2 input under income. 

Select Edit for this W-2 and scroll down to box 15, State & Local Taxes.  Confirm your input per state as reported on your W-2 Forms.

Go ahead and go through the state interview questions again.  MO should treat you as a nonresident even though you were a part-year resident.