All sort of issues here - while there's a short list of things that one normally looks for with extreme short cycling (such as a blown bladder in the pressure tank) that doesn't touch the 90 PSI issue...
If, in fact, water is going back down the well, the check valve in the pump (there is pretty much always one built into deep well pumps) is bad. Depending how your system is set up, there may or may not be another one inside the house, which would also have to be bad. My own system, after considerable reading and opinion-gathering, has only the check valve on the pump 300 feet down my well. When it goes, it's pump-pulling time. That would be the case even if there were others, and others might actually cause that to happen sooner according to a number of experienced well folks, which is why the other one I bought is still sitting in a box, not installed. In another few years I'll put a new pump on the shelf for when this one dies, so it's not a panic buy when (not if) I need it.
80-90 PSI is VERY high and almost certainly indicates a problem with the pressure switch. Certainly on my well the emergency relief valve lets go at 100PSI. On 40 off 60 is usually the upper range of domestic well pump "common settings" (30/50 and 20/40 being the other two commonly seen.) Ideally you'd know what your setting usually is or it may be recorded on a sticker somewhere on the system. You mention it going to the very high pressure when first switched on after being off for a while - what was it cycling between when you found it?
You could TRY hooking up a garden hose and opening that valve, then turning the pump on to flush out the crud that's been stirred up. It's possible-but-not-likely that this MIGHT clear a bit of crud that is jamming the check valve open. If nothing else it will get the stirred-up crud out of the well before the pump is repaired or replaced.
If your well is prone to throwing dirt, you may want to add a "spin-down filter" (I put mine after the relief, but before the rest of the system - this was contrary to the manufacturers instructions - they suggested after the pressure tank - but in line with the whole point of having it.)
The Pressure gauge reading cycles between the pump cut-in and cut-out pressures. Sounds like the gauge is dead and permanently stuck at 24psi.
Often, the cut-in and cut-out pressures will be listed on the pressure switch and commonly are 20psi cut-in and 40psi cut-out. So your water pressure on the gauge will cycle between 20 and 40psi during water usage.
The tank has a built-in bladder that is pre-charged to 30psi when your gauge reading is 0psi (pump breaker turned off, water tank drained). There should be a faucet somewhere on the pump manifold system that allows you to drain the pressure tank of water so you can check the bladder air pressure.
To repair the gauge, I would switch the power to the pump off, drain the tank and simultaneously, replace the gauge and check the bladder pressure and top it off if needed. Then power the pump back up.
If there is a lever on the side of the pressure switch (image below), it is a low pressure cutout to prevent the pump from operating if the well runs dry. Pull it up till the pump turns on and hold till the pump stays on (about 10-15psi). With no low pressure cutout, just power the pump on and it will automatically do the rest.
Once you have a working gauge, you can start to determine whether it is a pump pressure supply issue, or if you have something more serious like a restriction in the house supply line or a leak.
Do you have any sort of sedimentation filter in your water system? Sounds like it may have clogged and if a screen, needs flushed or replaced if it's one of those string filters.
Best Answer
Allowing for the possibility that it could be otherwise as diagnosis via internet is imperfect....
I see both a pump problem and a check valve problem here. While they may be located in the same place (if your only check valve is on the pump - which is the way mine is set up) they are not the same thing...though I also see another possibility that would do both.
The third possibility - you have a leak in the pipe above the pump and check valve - so the pressure is limited, and the water drains away from the leak without the check valve being to blame. If you can go and listen at the top of the well while someone else turns the pump on, you might actually hear this leak, if its in the well above waterline. Or you might have a recently soggy spot in the yard if it's after the pipe leaves the well.
While I don't think it's to blame in this problem, I'd also suggest rechecking your tank pre-charge to see if it has stayed at 38 PSI or started to fall again. Often when the precharge becomes low it's not simply a matter of needing to add air - it's a failure in the bladder or diaphragm.