You'll be travelling late in the season - be especially careful in New Mexico and Arizona in the higher elevations. Plan for cold weather - esp. between Socorro and Safford. Weather data is available at:
If the weather is really bad in New Mexico, I'd detour south thru Las Cruces and Silver City.
With average Dec lows in the low 20s (That's minus 6 for you Canadians), you should be prepared for some nights in the low teens (minus 12). You should also budget for motels. They should have cheap off-season rates in the small towns. Of course, you are from that "Igloo Country" up nawth - so it shouldn't be anything. Still, your sleeping bag should be 4-season.
This is just a guess - -
There are probably better county roads east of the Mississippi, but you would have to get county maps.
Go to the departments of transportation for each state and check traffic volume data:
AL - Co28, 169, 26, Co 2, 223, 29, 10
MS - 18, 540, 43, 28, 553
Natchez rhymes with "matches" not "sashay". Great place to cross the Mississippi. Bridge busy.
LA - 131, 15, 565, 84, 124, 459, 8 118, 171, 474, 191, 6.
TX - 21, 294, 287, 22, 6, 380
NM - 380, 1, 60, 12, 180, 78
AZ - 78, 191, 70, 60
Although this route roughly parallels US 84, I would avoid it as much as possible.
State parks and national forest campsites may be closed for the season. You can probably get away with camping anyway, but you could also be asked to leave - - I was in one late fall trip. Small West Texas towns are really friendly - they may think you are crazy - but you'll usually be able to camp in the town park - they may even put you up in the church basement.
Best - J