They're, in a word, bullshit, especially when we're talking about Austin. Texas is not a monolithic red sea. There are few things as tiresome on DU as the red state/blue state, lump-everything-all-together nonsense. Much of "blue" Minnesota, for example, is way redder than the big metros in Texas.
Yes, the state-level political leadership is nuts, but Austin is blue, as are many of the suburbs. I live just north of Austin proper and our district went about 60/40 Biden. All of our local politicians and judges are Democrats.
Been here for about 10 years and like it quite well. It's scenic, there's lots to do, tons of great restaurants and entertainment, heavy focus on health, trails, parks, and people are generally quite friendly (way more friendly than when I lived in SD and NE -- Midwest Nice, my ass).
The downsides are basically traffic, the never-ending heat, and housing prices, though coming from Boston, the latter probably isn't as much of a shock as coming from a lot of other places. Or the former, for that matter.
In my experience, people here also aren't as politically or religiously overt as the stereotypes would suggest. There are exceptions, of course, but you'll get that just about anywhere.