Fort Stevens is a good Oregon State Parks campground that's not too far away:
http://www.oregonstateparks.org/Haven't been there with our new TT yet but I tent camped there years ago and we liked it. It does get pretty busy in the summer though.
Alternative would be to go to the Washington side of the Columbia and try out one of their parks. I haven't camped at any of them yet but I've driven through Cape Disappointment and it seemed nice enough.
http://www.parks.wa.gov/parks/region/?selectedregion=pacificcoastEither way, it's not too long a drive to Astoria and you'll want to make a reservation as soon as possible.
The Long Beach peninsula is worth a visit if you have time. There's something like 29 miles of sand beach that you can drive on. And the food at
Jimella and Nancy's Market & CafeJimella & Nancy's Market & Cafe is literally 5 star class.
Hope this helps.
Chris