Winter in New Orleans is cold & damp. Many complain about “bone chilling weather”.
New Orleans is hot & humid, very humid, in the Summer.
Plus, I would exercise caution about living in a formerly flooded building due to hidden mold concerns.
“Southwestern” often refers to the weather & terrain, not necessarily to connections to “local/native cultural heritage and roots”. Although, if a cowboy hat is involved, it’s either Western or Southwestern or a fast talking used car dealer named “Honest Bob”.
P.S. I just googled the average temperatures in Austin, Texas. August has the highest temps, followed by July, then June, and finally September. Seems as though if one was out of town during July & August, then the weather would be fine. Is this not accurate ? Any complaints about humidity ? Seems much cooler than Phoenix, Arizona.