I thought I saw a tornado once, but it turned out to be a bunch of leaves.
The only thing that can keep you awake at night are heavy thunderstorms. Even then, they aren't as violent as the ones found in Dallas, but they generally dump a lot more water on us than up there.
As Mac said, the tornados here are generally weak. I remember people freaking out last year in Galveston when they had multiple waterspouts forming. The local news was all over that one. One did some damage to a car dealership, but were otherwise really puny (as far as tornados go). F-0 or at most F-1.