Besides running the well dry, there are a couple of things that will make a low pressure cut off trip. You can have too much air in the tank, or not enough air. If the bladder in the tank hits the bottom before the pump starts, it trips the low pressure cut off. Also might have a suction leak. After it sits a while it leaks back, loses prime, and can't stay above the low pressure kill point. This same problem is probably what melted your first impeller, and it will do it again without the low pressure switch.
If going in the crawlspace to reset the switch is a problem, you might want one of our EPS15/99 digital pressure switch. There is no lever on the side of this switch. You just cycle the breaker on and off to reset it.