There appears to be a real simple general rule of thumb, the southern state you go to, is the one that you can jump on the nearest expressway from. So you get I95, I75, I35, I25, I15, and I5. You'll find out that most southern states will have snowbirds that can jump on the closet of those expressways to get south. So in Texas you'll get MN, WI, OK, MO, etc... mostly. Florida will be MI, OH, PA, NY, MA, NJ, etc...
With that said, I think you might enjoy a trip to Texas. I would take the Trace Parkway down from Michigan. After that go down the Texas Coast staying a week here and there all the way to Corpus Christi.
I would head over to San Antonio from there (lots of traffic though) for some tourist stuff. From there head up into the Texas Hill country before heading back to Michigan. The problem with this route is you should take 4 months, not 2. Depending on those 2 months you could run into some very chilly temps in Texas.
Given only two months I would hop onto I75 and race to Florida hugging the Gulf. Keep going south until you find a spot that you can afford, and is warm enough for your liking. Good parks will be full if you plan on long term, which means you'll be in a sardine park, or short term public parks, or high price transit spots in nice private parks.
Two months really isn't long enough IMO. In what you'll spend in fuel you could add a little more $$$ and rent a condo and drive a fuel efficient car down. Florida is set up for that type of snow birding really well.
Good luck.
Thereโs no fool, like an old fool.