You must file for name change in the county of which your permanent address is. The current Fee: $301.50 ($83 filing fee, $10 administrative cost, $203.50 recording fee, $5 per certified copy). Make sure you sign the Petition for Name Change with CURRENT LEGAL NAME or it will be rejected. The Petition No. (in top right corner) will be filled in at time of filing the petition by the clerk, so don't worry about that. If you're filling out petition for waiver of court fees (MOTION AND DECLARATION FOR WAIVER OF CIVIL FEES AND SURCHARGES) for a name change, under section 1.2 write “PETITION FOR NAME CHANGE (Individual)." You will receive a court date via mail or email, BE THERE. The Judge will ask you some questions, such as if you are changing your name for fraud, and why you are getting a name change. Saying you are changing your name to match your identity or because you dislike your old name is legally valid, it is up to you if you would like to disclose your trans status.
Find info here kingcounty.gov/courts/district-court/name-changes
Once your name change goes through, make sure you change ALL appropriate legal accounts and documents (you might need a certified copy for each of these). Remember: to change your birth certificate you have to follow the laws of where you were born/where your original birth certificate was issued, so laws may vary from WA laws. Here is a checklist to make sure you don’t have any trouble down the line (https://transequality.org/sites/default/files/docs/resources/ID-Documents-Overview.pdf)