Have you checked the gauge?
Is the switch connected correctly? It is easy to get them wrong.
Here are some tests you can do:
1. With the pump off (using the circuit breaker if necessary) and pressure in the tank, check the air pressure in the tank (probably a bladder tank). The pressure should be the same as the pressure on the water side. If not, the water gauge or the tire gauge is in error. If the water gauge is wrong, replace it before moving ahead.
2. Operate the pump to at least 40 psi with all water turned off. Presumably the switch will not turn off (per your problem statement).
3. If the pump doesn't shut off at 40 psi with a reliable gauge, stop the pump. Observe the gauge to see that it is not losing pressure as with a leak or a foot valve failure. If you lose pressure, fix that problem before proceeding.
4. Back off on the pressure switch (the highest nut when you remove the cover, counterclockwise). It should "click" at some point which should be the shutoff pressure (40 psi).
5. Turn on the circuit breaker. The pump should not run.
6. Run some water to drain while observing the pressure gauge. Note the pressure when the pump comes on. That should be about 20 psi if you have not adjusted the "differential" (nut nearest the base in the pressure switch) of the new switch.
7. Shut off the water and observe the pump and pressure. It should pump to shutoff at 40 psi.
8. Adjust the pressure switch safely within the capability of the pump.
9. Check the air pressure in your tank as follows.
a. Turn off the circuit breaker and run water until there is no flow.
b. Check the air pressure in the tank. It should be set at 2 psi LESS than TURN-ON setting of your pressure switch. Adjust the air pressure if necessary.
Most pumps should be able to reach more than 44 psi if it is operating correctly and has enough water in the well. What make and model pump do you have? Is it a shallow well or deep well pump?