Since you can't do drop-offs, etc. you would not like 191 through Vernal, UT which would be 615 miles.
The easiest route for you would be from Cody, WY to 120 to Meeteetse to Thermopolis, WY then 20 south to Shoshoni; 26/789 to Riverton/Lander; 28 to 191 (not the steep part) to I-80 at Rock Springs, WY.
There's a KOA at Lyman, WY if you're ready to stop but going on to Heber City would be a more scenic stop.
Then take I-80 west to US40 toward Heber City, UT. This would be a great place to stay for one or more days. It's a pretty area and if you wanted to do so you could do a day trip to Park City, a former Olympic site or even into Salt Lake City. There's Jordanelle State Park, our favorite or Mountain Valley RV or River's Edge RV.
From Heber City, UT stay on 40 east to Duchesne, UT then 191 south (again, not the steep part) to US6 at Price, UT to Green River, UT at I-70. Then I-70 east to 191 and Moab, UT
This route is 605 miles through beautiful country - no drop-offs or steep highways and all are good for RVs.