Seriously, Rabs, as someone who has planned many-a-trip for long distances, if I were to do yours just to simply take in (besides the drive itself) some of the sites along the way, I’d need (at minimum) three weeks to a month to do what you’re attempting (and in fair weather, mind you). One of the longest trips I did (with my parents rotating driving) was from Nashville to San Diego, and with reasonable stops (and about 4 days at La Jolla), the trip was still 4 weeks in duration. At 11-12 days, with that route (and even if with the northern leg lopped off), you’d be in constant driving mode out to the coast with little time to do more than eat, rest and sleep. You’d just be better off flying, and it’d probably be cheaper, too. Sorry... :-(
I did DC to Yellowstone via Nashville and St. Louis in May in under two weeks. Actually, I showed up during the floods, too. And I definitely got to see Yellowstone, the Badlands, Rushmore, Little Bighorn, and Chattanooga during that trip.