Your idea of going to Death Valley first is a very good one. It gets hot beginning in March. Stovepipe Wells does have a row of full hookups and also Furnace Creek does, too, if you need the air conditioner. However, you might luck out going that early. Mesquite campground (dry) is very nice and it's near Scotty's Castle for the tour - a must.
You mentioned Carlsbad Caverns and another beautiful one east of Tucson off I-10 near Benson is Kartchner Caverns State Park. It's a wet cave and growing as opposed to Carlsbad. Both are great but in different ways. Kartchner has a lovely campground, too. For Carlsbad we stayed at Brantley Lake State Park.
For visiting Tucson and Saguaro Nat'l Park, staying at either Catalina State Park or Gilbert Ray campground at Tucson Mountain Park are in gorgeous Sonoran desert surroundings. Both have electric. The flowers will be beautiful in March.
Don't know if you're hikers but we made it to the top of Guadalupe Peak. It's quite a climb. Camping there is dry. Between El Paso and Guadalupe is Hueco Tanks State Park for some nice Indian ruins. They have a campground, too.
For Joshua do stay in the large boulder-strewn campground on the north side - not recalling the name right now. It's very unique.
For Big Bend we really enjoy the Rio Grande Village campground. It's a neat area. There's also a row of full hookups next door if it's too hot but the other campground has lots of trees for shade. We've also stayed at Cottonwood campground by Santa Elena Canyon but generators are not allowed there. You would have no problem getting into the higher elevations campground - Chisos Mtn. which would be cooler.
Have a great trip!