in Pioneertown, Ca. Out in the desert but only maybe a couple of hours east of LA near Joshua Tree National Park. Its a gem.
Host some big bands and you can stay in the old terraced huts. Book early as there are only about eight. There is an old film set you can walk about in too.
Route 10, Oatman, Arizona. A fantstic road in and out. Old pit donkeys wander the streets and shops.
Route12 south Utah as mentioned is a topper for miles and miles. TBH Utah is one of the most spectacular places I have been to. Buy your National Park pass before you go as its a lot cheaper if you are visting multiple parks.
Lee Vining, Ca. on the east side of Yosemite is small and fine but most importantly gets you next to Mono Lake and its salt structures (famous Pink Floyd photo from there) and an early start to head over the Tioga Pass 10,000ft and the park is pretty empty.
Stayed here a couple of times: https://lakeviewlodgeyosemite.com/
The 2nd time was after a 700+ drive from North Rim Grand Canyon (huts are lovely, sunsets & star gazing and the Canyon itself are tremendous) but the driving is easy over there and out in the sticks, little traffic.
3 weeks is a excellent amount of time so you will see plenty. Stay off the motorways after you leave LA. Don't miss State parks as can be just as impressive as the well known National Parks.
If we had any money we be over there every year. Started planning for 2021.