If you do not have a person who actually lives in WA to help you out. I would advise on not even bothering.

You can obtain a utility bill in your own name using just a Tax ID Number.

I obtained a Comcast Bill with just my name (it arrived in under 7 days of signing up)
Since I was not at the address I needed to write a letter giving written permission for the adult present (my friend) to have the service installed on my behalf + provide them with a copy of my passport (or any valid ID).

The 1st month service bill + installation was around $45.

For proof of identity bring:
Matricula ID Card
Foreign Passport
High School Transcript (Needs to be Official - Sealed in the envelope your school provided).
Official copy of your birth certificate, can not be a photocopy.

You take the knowledge test the same day it's 25 question and you need to get 20 right. You can also apply for a Washington state ID on the first day.

Then depending on if you passes the knowledge test you can schedule an appointment for the driving test which takes on average 7 calendar days.

For the driving test the car only needs to be insured and registered in the state of WA under the owners name, not yours.

If you pass everything you get a temporary license the same day, while your real drivers license card is processed and mailed to you within 30 days.

That section on their website that talks about having to sign a form if you don't have a Social Security Number is just there to scare out of state people from trying to get a license there. When you apply they never bring it up.

The state is fully aware that they are providing a drivers license to undocumented immigrants, and want to keep it that way.