I lived in central Florida for 40-plus years without a hurricane until 2004 when we got three in a couple of months. Years without the incident certainly doesn't mean the incident won't happen.
Don't just focus on tornadoes; the chances of you getting nailed by a specific tornado are pretty slim. And serious outbreaks don't happen real often.
Remember that the storms that time of year are caused by cold fronts which sweep across the southeastern part of the continent with violent weather. In addition to tornadoes you can get strong straight-line winds from downdrafts and big hail, either of which will destroy an RV (or car, house etc).
Personally I'd be elsewhere until mid to late May. Not that I'd be terrified and quaking in my boots but I'd just prefer to avoid that concern.
But if you must: Weather Underground wunderground.com has some good radar pages; I like to use their Wundermap for a general overview and the Nexad radar quick radar scans. The Nexrad page can display hail and vortex signatures frequently early enough to make it to the shelter.
A weather radio that will alert on SAME will wake you up in the middle of the night; beware that you have to program it for the area you're in. You can find that info at http://www.nws.noaa.gov/nwr/ Area listings are at the bottom of the page.
I've found an Android app called Radar Now to be useful while traveling. It's locator dot moves along with you while showing radars. Weather Underground's app is good but your location is stationary. These are on the Google Store.