I suggest you stop by 4 or so before the crowds, or, make reservations. That's about 400 miles a day. There is a KOA at Williams you could stay Sunday eve before you go into the park. Elk city OK, and Clinton OK have route 66 museums, Amarillo RV dealer has an interesting RV museum. Your not going to be stopping much on your way there. Traveling at 60 MPH with fuel food 400- 450 is a nice comfortable drive. Not a lot in New Mexico, open space and lots of it. Get a KOA value card if your going to spend more than a few nights, get 10% discount and earn points toward future stays.
Have fun and enjoy the country !
2011 Toyota Tundra double cab