Have you looked at the state parks - both Washington and Oregon? The coastal ones remain open well into the fall, often year around. Clallam county also has parks on the Olympic Penn, including Salt Creek. Cape Flattery at the far NW corner probably is still open, since it is on Indian land.
http://www.clallam.net/Parks/SaltCreek.html
http://northolympic.com/files/capeflatterytrail/index.html
Olympic NP has only 2, maybe 3, campgrounds for car campers. Most of the camping on the Wash and OR coast is state parks and private.
If I recall correctly Crater Lake is the only Nat Park in Oregon. There is a lot of Nat Forest in the state, but their campgrounds tend to be more primitive than most RVnet campers like. Plus most will already have been closed for the season by the concessionaires. The Federal closures shouldn't put too much of a crimp on coast recreation opportunities.
Now the California Redwoods might be affected more. Some are in state parks, but they are operated in cooperation with the larger Nat Parks in the area.