I think you have probably diagnosed the problem correctly, the backflow valve in the tank where the supply line from the pump attaches is probably stuck or even missing. REMOVE the drain valve on the bottom of the tank don't just open it!!! If there is even a small amount of pressure left in the tank when you remove that line it could be very dangerous, REMOVE the drain valve-it's easy.
Remove the line from the pump where it attaches to the tank then remove and check the valve (if it's even there!), if it is leaking or missing you will need to replace it.