The beauty of being retired is that you can go camping when the kids are in school and most everyone else is not on vacation. That being said, in my opinion both spring and fall are great on the Washington coast ( where we are currently). As stated earlier, fall tends to be dryer than does spring up here. I find that on the Washington coast, unless a big front is moving through, showers come and go as the clouds pass through, and then I go outside/down to the beach again. I love this place.
I love camping in eastern Washington, east of the Cascades, in spring and fall. The temps are down as is the threat of wild fires/forest fires. We avoid east of the mountains in the summer for those reasons, but spring and fall can be wonderful. Western Washington, west of the Cascades, can be a ****-shoot in spring/fall because the clouds and rain can back up to the mountains for weeks at a time and bring rain. You can never tell.