The pressure switch on on well system started leaking recently so I replaced it. I noticed that both the old and new switch were clicking on and off fairly rapidly (every 10-15 seconds when a faucet open). Knocking on my pressure tank it sounded like it was full of water and I got zero air pressure on it.

I shut the pump down and drained the pressure tank and pressurized it to 38 lb (system has a 40-60 pressure switch). I checked the air pressure several times to confirm that 38 lb. reading.

Then I turned the pump power back on and everything seemed normal. I rechecked the air pressure and it was now 54.5 lb. I drained that down a bit to 51 before I decided I should ask here before doing anything further.

My question is, should I bleed the air pressure off until the pressure tank is at 38 lb. with the well powered on? Any idea why the big jump in the reading?

The tank will only be at 38# when it has been drained. Once the system is turned on, the tank pressure will go up and down relative to the system pressure as indicated by your pressure gauge. So you should not adjust the pressure after setting it when drained as you did to begin with.

However, if this is a bladder tank, and it sounds like it is, then the fact that it was full of water is proof-positive that the bladder is ruptured. It will continue to operate OK for a few weeks until the air is dissolved and it's full of water again. Replacement is the proper course of action.