How much pavement? It is impossible to do entire trip on dirt, mostly on dirt, not all. How rough of dirt roads/trails do you want? Fast rally stage type of roads or Forest Service fire roads or pavement to off road trails? How much low range work and scratched skid plates do you want? How many days trip one way? Return to Tucson same route?
There are so many different routes to go. The east side and center of the state offer many ways north to south. Many of which are not possible right now due to annual and weather related closures. Keep in mind Indian Reservations, many do not like interlopers without permits except on paved roads.
If you can get reservations, Casner Mountain Trail is a fantastic trail. It runs between the Sycamore Canyon Wilderness and Red Rock-Secret Mountain Wilderness. Only a limited amount of fullsize rigs are allowed so permits are required. It is a cool way from Cottonwood(ish) through Parks to the Grand Canyon.
You can stay on the big roads and make it from Tucson to GC in about 6 hours. Throw too much play roads into the route and it will not be a quick day trip.
Get all the National Forest Motor Vehicle Use (MVUM) maps you can for Az. I think only Kaibab, Coconino and Precott National Forests have them at this time. They are the official "where you can drive" document.
You are welcome to PM me.