Perhaps you have an air leak?
I would test the system like this:
First, wait for the pump to run. As soon as it shuts off, turn off power to the pump (to prevent it from turning back on during testing). Open a faucet right away and observe the water pressure.
Turn the power back on, wait for the pump to run, then turn off the power to the pump as soon as it quits. Wait 15 seconds (or so...time the pump cycle to be sure) and open the same faucet. Observe the water pressure. Is it considerably less? Is it the same? If less, then the pump is probably doing it's job and a new pump won't solve the problem. The problem is maintaining pressure (indicating a leak of some sort), not creating pressure (indicating a pump problem).
So now you have a visual baseline for the pump cycle at high pressure (immediately after the pump stops) and low pressure (when it would normally turn on).
Now let the pump run but wait 60 seconds, then 120 seconds before testing the pressure. What's happening? If the pressure further decreases as the time increases between shut off and pressure test, that would confirm that there is a leak somewhere.