Georgia is not that much colder then Florida. Yes the occasionally get some ice but that's rare. Toyota Corolla's are generally considered reliable, and without a ton of financial details here it doesn't sound like it would be a great idea to buy a car at the moment. Don't use the move to try and justify a new car that you want because of hills and cold.

I also don't buy the same car is $5-10k less expensive over a state line. You are missing something or the people advertising these prices won't actually sell for that price. If the price is true when the time comes to get the car you drive to get it for that much savings. I have done multistate road trips to get cars before, it can be fun.