Both check valves have to be leaking for the tank to lose water and the pressure to drop. And yes adding a new one up top will probably stop the pressure from dropping after the pump shuts off. But it IS like putting your finger over a straw full of ice tea. The water is held in the pipe by a negative pressure. So when the pump comes on, it only has to produce a thimble full of water to change the pressure in the pipe from negative to positive. The resulting shock wave will hit the new check up top like a Prize Fighter hitting you right in the nose. You will hear and feel a "thump" when the pump starts. This water hammer can split pipe, break fittings, even shatter the thrust bearing in the motor.
I would try to remove the upper check by the tank, or at least remove it's guts so it cannot work. Hopefully the extra pressure exerted on the lower check will help it seat better and you won't lose pressure when the pump shuts off. This is the way a system should be set up (with the only check valve on the pump itself) as it will cause the pump to start against a positive pressure, which is a good thing.
If the lower check still leaks, you can add a new check up top, and see how long you can live with the water hammer. But if the check on the pump will not hold, the only real fix is to pull the pump and replace it.
