I've been taking out my '82 E-350 class C motorhome for years to camp in pretty much the same locations. Being from Rock Springs, we like to camp where there's trees, not too picky about being near water. We like to dry-camp rather than be in RV parks or Forest Service campgrounds (they never have any firewood). We've mainly been going to the SW corner of Flaming Gorge near Manila, UT or up to South Pass / Louis Lake area to camp.
Any suggestions for someplace new would be welcomed. Keep the suggestions within about 150 miles of Rock Springs, please. Gotta be a place a 28' land yacht can get to also.
Try Green Mountain near Muddy Gap. Drive to Rawlins then go north to Muddy Gap, then turn left at the gas station and go about 17 miles (towards Jeffery City) to the Green Mountain sign. Turn left there. The mountain is about 10-12 miles down a dirt road but you can see it from the highway. There are two campgrounds, one is free with no amenities except a bathroom and the other will cost you. They have water, toilets, etc there. Since you aren't interested in a campground however just keep driving to the top and there are many places to camp up there. There are several roads up top to explore. In the mornings and evenings, if you aren't too noisy and active you will likely see some mule deer and maybe some elk if you are in the right spot. The mountain is only open from mid-May to mid-November.
Don't know if turning at Bairoil will help you. I am sure if you take the right (dirt) roads you could work your way around to the road up the mountain. Never tried it myself. We always came from Casper so we turned at Muddy Gap. We (me, my dad and my son) hunted elk up there until this year. My dad died a couple months ago and I just don't feel like it. Anyway, we drove a lot of the dirt roads around the mountain but never really got to Bairoil.
Max elevation is about 9000, maybe 9500 feet. I never checked it. It is just sage brush hills until you actually start climbing up the mountain then it is pines and aspens. Both campgrounds have some trees although the fee campground has more. The whole mountain has trees except for the meadows, some of which are huge. You can expect to see marmots, deer, elk, antelope, wild horses and cattle along with other smaller wildlife.
There are a lot of roads you can explore, both on the mountain and at the base. The mountain is surrounded some BLM land but it mostly private property. All the roads are public for the most part. Enjoy it if you go.