I wouldn't worry to much. With 10 days you'll only scratch the surface of the desert southwest. Just look at it as a recon tour with blurry scenery. When you're older like me you can get a golden eagle pass and go back for a closer look.
With 10 days, out and back on freeways is the ticket to more time at the national parks. Mid May is a great time to ride the southwest. Less crowds. The north rim of the Grand Canyon should be open by then. If not, hit the South rim.