As several people said, there are some nice state parks on the Oregon Coast. One place to consider going is the Oregon Dunes which is mostly National Forest Service land. I spent a couple days in a nice campground on a "non-motorized" section. No reservations but it seems that some schools start up in mid-August so if you hit it on a weekday in late August it's not too crowded and usually nice weather.
One thing to remember is that if the weather is too cold and wet on the coast you can usually find nicer weather by going inland so it helps to be a bit flexible. When I lived in Oregon, I usually planned a camping trip the weekend after Labor Day up in the Cascades and almost always had nice weather although it starts to get cold at night in the mountains.
One thing you should take into consideration is that there is a solar
eclipse that passes right through the middle of Oregon on August 21 this year. Every state park in the path is completely booked so not a good place to avoid crowds but would probably be fun and educational for the kids.