Why did Washington State have me file a state return if Washington doesn't do state returns and why was I accepted after filing my state return

Although Washington doesn't have a personal income tax, TurboTax is supporting the filing of the claim for the Working Families Tax Credit.

The Working Families Tax Credit is a refundable tax credit for eligible low to moderate income individuals and families. To claim this credit, Washington is requiring residents to file a state tax return.

How do I claim the 965 then? And how soon can I receive it?

If you've already filed the return through TurboTax, you've already applied for the credit. Washington can take up to 90 days to process the refund. You can check the status by clicking here.