If your expansion tank bladder has failed, this would result in your pump operating any time you were consuming water. Your pump would cycle every time you opened a faucet or flushed a toilet. It would shut off when you closed the faucet or when the toilet finished filling and the cut-out pressure was achieved.
It appears that your pressure switch may not be working properly, therefore letting your pump to continually run until it trips its thermal motor protection. After the pump sits idle them the thermal overload resets, but the pump should only start if the pressure switch has achieved the cut-in setting. You should NEVER need to manually close the contactors on a properly functioning pressure switch.
Have your system serviced and repaired by a competent and licensed contractor. Sounds to me like you need a new expansion tank and a new pressure switch. Good luck.