Tornadoes aren't going to be the problem. It will be the Heat & Humidity in late Spring through very early Fall. And you get that weather all the way up to the Mid Atlantic.
I thought this past Fall sort of sucked. First part was to much like early summer, and now it feels like winter. We barely had two weeks of fall like weather.
One thing about the south is the closer you get to the coast it won't be to bad in the winter.
And don't be surprised if some places in the south east are just as expensive as the north east.
I see someplace in Florida working. Doesn't sound like lots of options.