Knowledge is often a way to overcome fear, or at least manage fear. If you don't have a weather radio that broadcasts watches and warnings, you ought to buy one and use it. You might also want to check out our thunderstorm and tornado pages, which have links to information that will help you understand storms. I've never seen anything in tornado safety information about taking refuge in a crawl space. But, I wouldn't do it.
Tornadoes often shift houses on their foundations, maybe only a few feet.
Imagine between being between the beams in a crawl space with only a foot or two of space between the ground and the floor when a tornado shifts the house a couple of feet. I think you'd be better off in an inside room, away from windows. Under a strong table in an inside room would be best. Another concern, I grew up in Florida where crawl spaces are much more common than basements. There was a general belief that rattlesnakes loved crawl spaces.