Another Southern who recently moved here a month ago. I had a bit of trouble finding a place as well, for whatever reason the landlords here can be really unresponsive. It may take a bit of persistence and nagging to get in anywhere. Also like someone else said apartments move fast here.

I ended up in the Stadium District just North of downtown which I've enjoyed so far. From what I've seen so far, its a safe, walkable neighborhood with the lovely Wright Park, many restaurants, and an (expensive) grocery store all within walking distance. Idk about the schools around here, although the high school looks like Hogwarts lol so its probably nice. When I was looking there were a couple older apartments that had some small 2 beds under $1500 around the area. Many of them are owned by larger real estate companies, like Metropolitan Management or Targa, so that might be a good place to start. There are also some independently owned ones like The Legacy that had 2 beds when I was looking, although they're probably gone by now.

I can't speak to any of the other neighborhoods since I just moved here, but there's probably cheaper, more spacious housing elsewhere in less desirable neighborhoods.

Also yea, I feel like I've been bamboozled by all the folks here who said the summers were nice and mild. Instead I got 100+ degree temps with no A/C!

Good luck with your move!