Michael, not sure if you live east or west of the Cascades; we live east so will start there. If you want others west - let me know.
In eastern Washington, our favorite place to camp (close to home) is Charbonneau Park on the Snake River about 20 miles east of Pasco. It is managed by the Army Corps of Engineers and sites can be reserved through ReserveUSA (
http://www.reserveusa.com). The sites are big and have lots of shade trees. Bathhouse is nice with free showers. Not all sites have full hookups, but all have electricty (30 amp). There are several sites there that are non-reservable. I think if you come mid-week that would be no problem. I also have had good luck surfing the reserveusa site for availability at last minute and have lucked upon campsites that apparently were cancellations.
There are also lesser used campsites on the Snake River and the Columbia. Will provide more detail if that's your interest.
Regards,