The reason that neither Washington nor Texas is on the list "to choose from for working out of state" is because neither state has a personal income tax.
So there's no state tax return to file in either state.
You will probably want to at least look at filing at any state that was mentioned on your W-2.
Note: if Washington and Texas are mentioned on the W-2, it will be only the state code because there won't be any dollar amounts. Just ignore them.
