I certainly see people with Class Cs in BLM lands along with 40' DPs so I don't think you're going to have a problem with a C. A person with a 4X4 pulling a TT may get to more places than you will with a C, but then you don't want a 4X4 anyways.
We have hydraulic lifts on our DP and we camp on more dirt than on paved or cement pads. We are probably double what you'll weigh - you won't have a problem. Get some jack pads for the softer areas you might find. They'll be handy for longer stays in case it rains.
You didn't mention looking at a Class A, but those large front windows are sure nice - like a picture windows in a house.
What did concern me a little was you're comment about heading north to the snow. While I can certainly understand that after being in the heat, you may want to think more on camping in it.
Cold weather campingCold weather campingCold weather campingCold weather campingIf work allows, I'd stay down south (try out west for a change) and up north and higher elevations (try out west for a change) to stay cool during the summer.
Bill