In the last few days, our water pressure has dropped dramatically. How can
I diagnose whether the pump has gone bad, or if something is partially
clogged?

I have a deep well with a submerged pump, and a 20 gallon bladder tank. The
pump runs always, and the pressure in the tank, according to the guage, is
just under 20 psi. I'm note sure what is used to be, but I think it used to
turn on at 30psi, and off at 50 psi. If I open a faucet, the pressure
reading drops to about 15 psi.

There is a valve in between the tank and the house. If I close that valve
(shutting off the water to the house) the tank still never gets above 20
psi. With the valve still shut, if I shut off the power to the pump, the
pressue slowly drops to zero. (It takes about 60 seconds.) It used to stay
pressurized.

I understand that I can check the impedance of the pump, and know how to do
that, but I haven't because I have no idea what it should be.

For what it's worth, our water has a lot of iron.