You will enjoy the lack of bugs and humidity once you are west of the Mississippi. Be sure to bring along a rain jacket and long sleeves for time out by the campfires. you will need it more as a wind breaker than a rain coat in most cases. You probably will not run into any snow, but the higher elevations can bring snow any time of the year.
Look at the maps of Yellowstone. The only full hookup campground is in the center of the park, flying bridge. It can have bears. West Yellowstone is the closest full hookup, and is slightly outside the park, but still less than 3 hours drive to Old Faithful. THe park is in a large figure 8, with campgrounds in all directions, Flying Bridge in the center of the 8, west Yellowstone to the left of the center of 8, others are at the top and bottom.
The bottom exit is "Near" Grand Tetons NP, but they are separated by a couple of miles. There are a couple of boondocking places in between the parks, but somewhat difficult to find. It is west of Flagg Ranch, another full hookup campground. There is only 1 road going west, it is on that one, just about where it changes from paved to dirt, about miles west of Flagg Ranch, there is camping areas on both sides of the road, about 2-8 miles from town (separated by about a mile or two). After driving 8 miles, there are no more camping areas.
When you leave Mt. Rushmore, turn to the right, go about 1/4 mile, there is a tiny place to get a side shot of Washington. My suggestion is let the co-pilot out to take the picture, then get moving again. Along highway 16, there are 8 tunnels that line up with Mt. Rushmore in such a way that the monument is straight in front of the tunnel. Stop at the second or third tunnel, less crowded than tunnel #1 and #4 (first from the other direction).
I stayed at Custer state park, and took my car on the wildlife dirt road. You can take your RV too - just stick to the first 3-4 miles of fine gravel road, then go back to town. If you have not seen any wildlife by that time, you are at the wrong time of day, an likely will not see anymore.
Have fun!
Fred.