I guess I sound like a broken record...check your pressure tank. It should sound hollow on one end, and make a duller thud on the other end. It is supposed to have air in one side.
It sound like it is shot. The pump runs until it reaches the high shut off point. Normally, in that process, it is not only replenishing the water that you used, but also compressing the air in the tank. Once it reaches the high point, the pump turns off. The air bladder then pushes the water from the tank to your faucet rather than the pump until the pressure drops to the turn on point, and the pump starts again. If there is no air in the tank, the water doesn't compress the air bladder, and the system reaches the high shut off point by expanding the pipes a tiny fraction. Since there is no air in the bladder, there is nothing to push the water out and what little expansion of the piping system there was creating the pressure, is relieved. YOu might get a small dribble out, but not much.
So, rap on the tank. If it sounds the same at both ends, shut off the pump, open a faucet, then use a tire pressure guage to check the pressure in the tank. You might get some water out, but probably not. It should be 1-2 pounds less than the low turn-on point of your well system. If it doesn't hold air pressure, that's your problem.