This is a better question for r/legaladvice.

I am not a lawyer this isn't legal advice, just an opinion.

If you notified your landlord in writing in accordance with RCW 59.18.575 you are likely not responsible, but you will need a lawyer.

If you didn't do it in writing, but the landlord told you to leave and that you could break your lease (hopefully in writing), you may have a case. I'd talk to a lawyer, but you may also want to review your lease to see the terms about lease termination. $12,000 sounds way too high for breaking a lease even without DV or permission, and if they knew you were leaving this could be an illegal charge.

You likely want to immediately dispute the debt and request validation/verification via certified mail with return receipt. There is a ticking clock on this, 30 days from when you received the communication, after that runs out you can be held responsible for the debt even if you have a case. Do not delay on this or you will have a huge debt you will not be able to get out of.

Talk to a lawyer immediately.