We did a similar drive when we moved here last summer (NJ-Tucson, AZ, but our 2nd stop was Atlanta). Anyway, from Atlanta to Tucson, we took a more Southern route than you are (I-10), but I'll tell you how long it took from Atlanta, and we stopped a lot.
We left Atlanta after dinner and drove to Montgomery, AL for the night. Left early morning for Mobile, spent the day there with friends, couldn't get a hotel room in Mobile (insane, I know, but true!), so we drove to New Orleans after dinner and spent 2 nights with my inlaws. From there we went to Houston, spent a day there with friends (from this board actually!), then left the next day for San Antonio. Spent the day there with family, drove to Kerrville and stayed there for the night. That was another hard to book place, we literally got the last room in the last hotel we called. We did a long haul day from (basically) San Antonio to Las Cruces, NM the next day, got to NM around, hmm, 3pm. We left Las Cruces the next morning and drove straight to Tucson, getting here around, maybe, 1pm.
So, 6 nights from Atlanta. We did about 6 hours in the car each day, then spent time in each city.