The planes are not the most comfortable, but they are not unsafe. The MEM-TUL legs are only about 340 miles, or about an hour flight time. Summer in the midwest often means storms, especially if you fly in the afternoon/early evening. Count on some turbulence associated with that.

I suspect no amount of convincing from anyone on this board is going to relieve you of your anxiety. That appears to be another issue. I agree, you may want to look at other arrangements.