I see Class A and CLASS C's towing enclosed trailers all the time. If I were in this situation, I'd want nothing else but an enclosed trailer for my car. Like you indicated, it provides much more storage space than just the car. You've got my vote for the enclosed trailer! Better for the car, wheels will not be rolling, no wear on joints, tires, suspension (except for minor bouncing in the trailer), wheel bearing, and BUGS! However, you do have 3 "vehicles" you'll need to register and title: CLASS C, car, and enclosed trailer (something to think about).
One thing you should check out before you purchase the trailer is the over all length of your complete "rig". Assuming you are not going with one of those multi-million dollar Class-C's that are as long as a train, you do need to be sure you're overall length (Class-C and attached trailer) does not exceed the length limitations of the state you register the vehicles in. That's the only "catch" with a trailer. Other than that, I think you're right on target with your plans!
FYI, every campground I've ever been to, has a spot to park second vehicles or utility trailers. Unload at your camping spot and move the trailer to the designated parking location, if your campsite does not have enough room.
On, one more thought: get a car with a hitch that matches your utility trailer and then you can move the utility trailer much easier without having to disrupt your campsite first to go fetch the trailer!