First, US 189 doesn't connect Jackson and Rawlings, US 287 does. If you follow US 189 from Jackson, you'll go SW to Kemmerer and eventually I-80 not too far from the Utah border. US 191 takes you south to Rock Springs, still far to the west of Rawlings.
If you travel Jackson to Rock Springs, US 191 is not a bad road, two wide lane hwy with a wide berm for most of the drive. There are grades along the way, but most have passing lanes so you don't hold up traffic. Just pay attention to your temperature gauge, if it starts to climb, downshift and let engine cool down.
From Jackson, head south on US 26/89/189/191/287 (!) At Hobart Junction take US 189/191 East along the Hobart River. At Daniel Junction, stay on US 191 to Rock Springs an I-80.
I prefer leaving Jackson/Grand Teton by way of US E 26/287, going over Togwottee Pass, through Dubois, stay on US 287 to Lander, and Rawlings. You'll pass Split Rock, a famous spot on the Oregon Trail and Pony Express route.
All these roads are very doable in an RV, wide lanes and berm, no narrow, twisting roads to give an RVer heartburn.
One source of information is Google Earth. On this you can select a section of highway and see a profile showing elevations. Another source is the Mountain Directory, a 'trucker's bible' of highways and their grades: