First, the bad news. The State of Washington charges tax payers to access their public lands. Water access points such as parking lots or beaches and recreation access points such as trail heads, day use of parks, viewpoints, and some roads in state forests require the purchase and display of a Discover Pass. The pass costs $10 per day, or $35 for a 365 day pass.
I've heard from several people that requirement of the Discover Pass is strongly enforced. They provide the yellow paper hangar when you buy the pass, but the instructions only state that the pass must be displayed. I've heard one story where a Fish & Wildlife official was going to ticket a person who had the pass on the dash in full view, but not suspended from their mirror. Sounds like somebody was having too much fun writing tickets.
AllStays or Freecampsites.net or the more complete (for public campgrounds) USCampgrounds.info so you'll have to do a bit of extra homework.
The Washington State Department of Natural Resources has a Recreation Guide that lists all of their campgrounds. It is available for download as a PDF, but the layout makes it a bit challenging to use on a computer screen. It can also be ordered or purchased as a printed version.
The Washington Department of Fish & Wildlife owns considerable land for wildlife management. On their website they provide tools (in the left hand column) to find out about public water access points and wildlife areas. Unfortunately, a random but fairly extensive search suggests camping is rarely (never?) allowed on the listed properties although their Public Conduct Rules allows for 21 day camping in a 30 day period. I just haven't yet been able to find the land where WDFW allows camping...
Next time we're in Washington, I'll spent more time leveraging that $35 investment.