You would, most likely, have an indication of a possible tornado or bad storm. You should be monitoring the weather with some kind of device.
You definitely should not keep driving or just sitting in your RV. You need to get into a shelter. You also should not park under an overpass. If a tornado should happen it would be twice as bad in that wind tunnel.
If you're in a RV park/campground, the best place is the restroom. There aren't many windows and it's usually a concrete block building. We spent many hours at various times in the Alabama state park restrooms - men, women, children, pets, lawn chairs, etc... they all went in the same one. The rangers came and knocked on doors alerting us to the bad storms.
When traveling in those states and checking into a campground, the first thing you should ask is where is the nearest shelter. Sometimes they're even marked on the campground map you are given.