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I would suggest that there really is not much of an off season in that part of the country. Rain hits most days, so really only cold weather is a factor. Seattle is known to experience 50 degree F/10 C in the middle of July, so really you have to be prepared for cold weather anytime.
Some high elevation campgrounds can be closed for the season due to snow. Mt Baker allow skiing until May, and highway 20 closes from early November until early May. Depending on storms snow chains could be required on highway 90, but they are probably provided with your rental car. However most campgrounds stay open year round, and there should be plenty between Seattle and Portland. You can find and reserve your camp spots using this website for Washington.
Using the website it looks like Millersylvania would be perfect for your needs. I've never camped there so cannot comment on the facilities, but most places are pretty darn nice in WA state.
If I was you I would push my trip out to the Olympic peninsula and drive the coast rather than I-5. I'd look to camp in places like Cape Disappointment and Fort Columbia. The beauty is stunning. You may have heard much about driving up the California coast, and the same geological formations continue into Oregon and Washington.
But reading your question you might want to push for the interior and hit places like Alder Lake and Mossyrock.
I would advise making reservations for your trip. Campground can sell out if a bout of nice weather hits the area.
My own experiences are mostly in the NW section of Washington state. I've frequented places like Spencer Spit, Rasar, Deception Pass, Baker Lake, Fort Ebey, and Fort Casey. Although this was about 20 years ago, I feel that they are just as amazing.