We drove from Rock Springs up to Jackson back in 2012. The only grade I can recall is just before Hobart Junction. Not sure whether I used lower gears going down, but I probably did, as I like to save the brakes.
This year at the end of May we drove from Casper to Lander using route 220 to Muddy Gap and then route 287 to Lander. A couple of days later we drove 287/26 over the Togwotee Pass to Jackson. I recall a pretty good downgrade (which you will be climbing) on 287 south of Lander, a very gradual climb between Dubois and the Pass and a steeper downgrade to Moran Junction. Nothing I would hesitate to do again. No experience driving 287 between Muddy Gap and Rawlins.
Personally I think that route 191 is boring. Routes 26 and 287 are a much more interesting drive. You could also make the short side trip to
Sinks Canyon State Park.