Roads are good in WY. Some are 2 lane but they are good. The only one to avoid is 14A. I have a 40'er and my route from Gillette to YNP is as follows:
1. I90 to Buffalo. We stay at Deer Park on 16 east side of Buffalo. Not fancy but ok.
2. 16 to 20 at Worland
3. Go thru Thermopolis to Shoshoni. I recommend some time in Thermopolis if you have it. We dry camp at Boysen State park south of town. Beautiful drive to Shoshoni.
4. Right on US26 in Shoshoni.
5. Right on US26 in the "big" town of Riverton.
6. Follow US26 to Moran Jct. and go north to YNP or south to Jackson Hole.
Now, you will be crossing Togwotee Pass which is steep. But it is not a problem. New road but there may be construction still.
Alternate route. Go to Cody on 14 and into YNP. There are some grades but not bad. When doing a grade, down shift to 2nd or 1st with a gasser. Kick on your retarder with a diesel and start slowly down the hill. Brake as needed.
I'll be in Jackson for most of the summer. Maybe I'll see ya there. Great spot for summer.