The BLM area you're referring to is known as Sunfair Dry Lake to the locals. Don't know why, because it is labled "Coyote Dry Lake" on the maps. It is a good place to camp, but the last part of the road in is dirt and can be rough. (Note: it has been several years since we have been out there, so info needs to be verified.) But there is plenty of room for any size rig, and finding a level spot is NO problem. Beware, it is a popular off-roading spot and on holiday weekends you could easily have a LOT of company!! Strictly dry-camping, no services. Closest full-size grocery store would be in 29 Palms.
Regarding campgrounds in Joshua Tree NP, the one at Indian Cove has sites that accomodate big rigs. Water is available at the Indian Cove ranger station, but may be difficult to access with your rig as it is a spigot on the side of the building. Reccomend you get water elsewhere if you're not confident maneuvering your rig in tight spaces. Several other campgrounds have sites that should accommodate your rig. Cottonwood Campground at the southern entrance of the park has some large sites and is a delightful place to camp.
There are basically three routes from the south to get to the 29 Palms area. First, the Cottonwood Springs road mentioned by Mutha, which is pretty much as described. We took our Class A that way, no problem. Then there is Hwy 62, which runs through the entire area. You can get to that directly from I-10 in the west just north of Palm Springs. It too will be a long, long, uphill drag going up through Morongo Valley and then tops out at Yucca Valley. Not nearly as scenic. The other way to get to Hwy 62 is via the road north from Desert Center on I-10. Not as steep, not as scenic, longer way to go, but a good road.
Other camping options in the area include the military campground at the 29 Palms Marine Base, an Elks lodge in 29 Palms (right on the highway and a good place to get water if you're heading for dry camping), and an Elks Lodge in Yucca Valley (Noisy!).
Hope this has answered your questions - enjoy your stay!
Bryan & Susan
2007 Pacifica TE-40B