Completely control your water samples by using a local water testing laboratory. The modern steel models are cheap and low maintenance and the diaphragm is set into them.
Here's my question: Is replacing the pressure switch the next reasonable step to take, and if so, is there any harm in going ahead and installing a switch since that seems to be what the pressure tank is set for? Don't forget to turn off the circuit breaker when you readjust the pressure switch.
On a building water pump-operated water supply system there is virtually always a pressure control switch somewhere; on some irrigation systems or other special designs the pump may operate only with an "off-on" switch and there may be no pressure tank and pressure control switch.
If you touch them, especially being near water piping, there may be a serious-risk of death by electrocution. I tend to always go the electrical panel anyway the find the correct breaker to the pump, from experience I always lean on the side of caution.
I would find PSI to be an intolerable water pressure, no wonder your wife is complaining!
If it is cycling, then you may need to adjust the air up or down. Add air to the pressure tank via the air-fill valve using the portable air pressure tank. The pressure tank sits adjacent to the well pump. After doing some research, I gather that 37 PSI is approximately where the pressure on the tank should be for a pressure switch; but if the cover of my switch is to be trusted, I have a switch installed.