When we turn on any faucet to draw water the pump goes on and off rapidly and repeatedly until the storage/holding tank is full. It recently started doing this. Prior to this it used to come on and stay on until the tank was full and then it would shut off.

What type of tank do you have, a painted bladder tank with an air valve, or a unpainted galvanized tank with no air valve. What does the pressure gauge read with the pump off?