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.
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