Lake Pleasant in Bothell is a nice park. I believe it is a contract park with Children's Hospital, The Seattle Cancer Care Center, and University of Washington Hospital for people who need to "camp" during treatment.
Another nice park that is near Seattle is:
Blue Sky RV Park in Preston, WA
https://www.blueskypreston.com/
This is just off of I90. Nice park around 60/night.
I stayed here while on a contract gig. Really convenient to Seattle/Bellevue area.
I grew up in Seattle as did the wife. We moved away 20 years ago and prefer not to even visit now. If it wasn't for the cancer center there we wouldn't even go there. The city is now filthy especially the downtown area. I'll leave it at that!
If you are thinking of a stopover to go out to the Olympic Pennisula area or a base to venture around Washington stay out of Seattle.
The state parks (on the West side of the Cascade Range) are booked heavily. You would have better luck in the Inland Northwest as a base camp for touring and as a stopover for going to the coastal areas west of Seattle.
Just my 2 cents.
You could also just hang at my place in July. Warning it can be hot!