I have lived most of my life in Oklahoma and Florida. I have been under hundreds (maybe thousands) of tornado watches, warnings, and alerts. I have yet to experience an actual tornado. In the scope of weather, tornadoes are geographically isolated, but very destructive event. I would not pack up a campsite just because there was a tornado watch. Usually, by the time a watch is issued, the watch would expire before you could escape the area.
Having said that, I would keep a close eye on the weather conditions in my immediate area. If I saw extremely low and angry looking clouds, or saw a funnel, or heard a noise like a freight train, I would look for shelter in this order:
1) a concrete block building with few or no windows. Sounds like most bathhouses I have seen.
2) a wooden structure with plumbing. And hang on to the pipes if you hear a train. Often, a wooden structure will be destroyed, but the pipes remain in place.
3) hang on to a water pipe at an empty site, and pray real hard.