As someone else has said, the answer is in what you believe you need to be a happy camper. If you do not need a swimming pool & full hookups, staying in the parks themselves would be the best experience, IMHO.
That said, here is an old post of one of our more recent visits to a National Park in New Mexico as well as a State Park (even a BLM Campground).
Santa Fe & Bandelier Nat'l Mon't area of New Mexico -A Trip Report 2012.
Los Alamos is a very neat small city in itself. Free bus system with very neat buses. Very pretty setting up in the high country. Somewhat of a gated community as well. We only drove through & stopped at a SuperMarket. A nice town.
http://www.visit.losalamos.com/http://www.losalamosnm.us/transit/Pages/default.aspxAs for having trouble getting into some of the high elevation National Parks, I wouldn't think you should have any problem learning to drive in the mountains. Main thing to remember is to keep your speed down & not allow the rig to get going too fast for any curves, especially on going down hill.
Oh, yes, in most of the western states, even in the hot southwest states, normally one does not to run the A/C at night to cool off. It usually cools down greatly at night out this way. Higher elevations & low humidity are factors. Therefore, one does need to have electrical hookups as one does back east.
In parks like Yellowstone & the Tetons, it is even less likely to need them. Maybe to keep warm would be nice at time however. Ha!
The last 2 visits to Yellowstone was in July (2010) & Aug (2009). We stayed at Mammoth CG in 2009 & Grant CG in 2010. We have been 4 times all together & never had a reservation. Madison, Canyon, Bridge Bay as well as Grant & Mammoth CGs all should be places to consider. However, you would have to check the park web site to make sure about the 37' rig. Mammoth has some site which would work. There were really large rigs there.
South entrance via the Tetons is the easiest route into as is the North entrance by Mammoth. West entrance would be easy enough, but take a lot more driving to get around to it. East entrances one will have to do some climbing.
Here's an older post with info from our past trips, etc.
DesertHawk's Recommendations for Yellowstone Area.