Moving Between States w/ Different Filing Regulations

YES...you must file an NC part-year tax return for all of the 2019 income you received during the time that you lived in NC

In the My Info section, where you Edit your name, you must indicate that you lived in WA on 31 Dec.....but that you moved FROM NC  and the date of the move.  That should trigger the NC Part-year resident tax return when you go to the State section of the software.

1) After getting the My Info section set up properly...make sure all of the Federal income and deductions& Credits is fully filled in first...every scrap....and reviewed and error checked.

2)  Then go to the State section and start the NC part-year resident tax return.   When going thru it's interview, the software will present all of your Federal income...dn have you indicate what sub-part you earned or received while you were an NC resident...if you moved in Dec, then most all of it will be NC income...with some exceptions for Dec interest or Dec mutual fund distributions. 

Your NC jobs W-2s will need to be tagged as NC PY-Source Income...but you'll see that selection as you step thru the NC interview..

Even though TurboTax didn't prompt you to file a North Carolina return, since you had North Carolina income you should add North Carolina and file a return. 

I would imagine that you also had North Carolina withholding in Box 17 on Form W-2 and you may be entitled to a refund of some of that withholding.  

Click on State in the left side of them menu in TurboTax and then Add a state to add North Carolina and walk through the interview questions to file your North Carolina return.