Jim,
Is your trailer a fold-out camper? Or is it just for carrying all the "stuff" you need? That will make a difference in what sort of campground you will be looking for, I presume. But in the meantime, let me give a couple of recommendations: In Glennallen (where I happen to be seated at the moment) Northern Nights campground is a nice one, with really nice people operating it. Cabins, RV parking, tent spaces, as well as showers. Not sure about on-site laundromat, but there is one about 3/4 mile west of the campground.
About 15 miles west is Tolsona Wilderness Campground. A beautiful site along Tolsona Creek - full facilities.
In Valdez Eagle's Rest is about the biggest, with everything, and just as you enter the main town. From there you can walk to just about everything in Valdez, and see the whole town.
As for the rest of the state - somebody else may be able to offer some suggestions. Living here, I don't usually need to camp so don't get acquainted with very many.
Jack (aka Alcan Rider)