Once you get into mid November you can get some pretty severe snowstorms in northern AZ and NM. You’ll minimize that if you stay below about 4000 in AZ. October’s great weather, high or low. The Phoenix area’s roughly 1100’ and you’ll be fine there in the winter. It can even get pretty cold if you’re away from the heat islands.
I camp high and haven’t done much in southern NM, but I think the terrain there, in general, is a little higher so you may get more chance of snow. But someone from NM can give you better input. I’d do as someone suggested, do your high country traveling early and move south as it gets later in the year. And the weather isn’t predictable, so keep up with it. Have a nice trip.