If you have flexibility in your travel plans, I suggest driving on a clear day when there is no snow in the forecast, and using the all-seasons.
A month of driving on the snows in the heat won't affect things much, but why put the wear on them if you don't have to? The rubber in the all seasons is not ideal for cold weather driving (it is harder than the soft rubber on the snows) but if you aren't driving through a storm, and you are making the trip down in one or two days, you should be fine. Just travel at the speed of traffic and leave a bigger gap than usual from the car in front of you.
According to Google, the drive will take about 30h each way (I did Flint to Tucson for a good guess), so I would break it in 3 for 10 hour days. By day 2 you will be well into warmer areas and will be fine.