Switchblades were removed from the state's "prohibited weapons" list in section 46.05 of the Texas penal code effective September 1, 2013, thanks to Rick Perry's signature on HB1862. Note that this legislation did not remove carry limitations on blade length or the carrying of Bowie knives, dirks, or daggers.
IANAL, but it's my layman's understanding that double-edged knives may be considered dirks or daggers by a court, so perhaps carrying double edged OTF - Out The Front - switchblades would be unwise.
The only local laws I'm aware of that may be problematic are in San Antonio - if anyone knows of others, I'd like to hear about them.
After the law became effective I got myself a couple of "conventional" layout utility-type folding automatic knives, and so far, I like them.