A Few More Things to Check Out
While I would also normally urge you to visit Death Valley on a RoadTrip that includes Las Vegas, the time of year you'll be traveling may present a problem. As Subman points out it can often be "the hottest place on the planet" and for that reason RV rental firms place restrictions on when/whether you can take their vehicles into the valley. I believe the restrictions normally apply to August, but you should check with your particular firm about your particular vehicle during your particular travel period. And just so you know, virtually all rentals in the US are equipped with tracking devices so they will know where you went.
One of (but not the only) reason that we recommend paper maps is the wealth of information they can present in synoptic view. Something that is simply outside the realm of electronic devices and their limited screen size and resolution. For planning purposes one incredibly worthwhile bit of information is that they often mark scenic roads, usually with a green dotted or dashed line parallel to the highway. Seeing all those possibilities laid out over a large swath of country can quickly lead you to roads that not only fulfill your desire "to see those rural, off the beaten path places" but also keep you going in more or less the right direction. The Federal Highway Administration has designated some of the best of these as America's Byways.
Some roads along your route that I've particularly enjoyed that you may want to consider for inclusion are: AZ-66 between Kingman and Seligman, the old US-66 (Route 66 of story and song) roadbed through the desert and the visual inspiration for the Pixar movie "Cars"; Oak Creek Canyon Scenic Drive from Flagstaff down to Sedona; the Turquoise and Salt River Trails which would make a nice loop from Albuquerque up to Santa Fe and back; and the Creole Nature Trail, a collection of roads through the Louisiana bayous.
AZBuck