If the lever you are referring to is on the side of your pressure switch:
Then I unfortunately have not very good news for you. As the name implies, this cuts off the switch entirely when the pressure drops to about 10psi below the cut-in point (eg, most well pumps are set up to turn on at 40 and off at 60psi, so the low-pressure point is about 30psi). It's there to prevent the pump from running dry.
Potential cause: low-yield well
Without any other information, my best guess would be your well is not producing enough water to keep up with demand. This seems to be consistent with your statement that it only happens after 30-45 minutes: there is enough reserve capacity in the well itself to supply you for a while, but ultimately, the rate of water entering the well is lower than the rate you're using water.
The ultimate fix for this is to install a cistern or holding tank that the well pump supplies, and have a second pump in that that actually supplies the house. The cistern is controlled via float (instead of pressure switch) and the well pump is protected by a load-monitoring device (eg: Pumptec by Franklin-Electric) that turns off the pump electronically if it senses no-load. The size of the tank largely depends on how much capacity you want, what the actual yield of the well is, and logically where the tank will go and how big you can fit. Commonly, the cistern is a large cylindrical PVC tank installed in the basement.
Potential cause: undersized pump
The other main cause would be an undersized or under-performing pump that is unable to keep up with demand. The symptom of this would be that the pump runs constantly during use, rather than cycling on and off, and that you generally have a consistently low pressure for that time prior to the low-pressure switch cutting out.
Replacing the pump in this case may help, but if you have a low-yield well then it's a waste of money. Considering it's been happening for 14 years, it seems unlikely this would be something mechanically wrong (I'd expect the condition to worsen over time, or for the pump to have failed by now) and more likely that the pump was undersized, IF you also have the symptoms I described.
What next?
If you can determine the pump is undersized, then replacing it with a larger pump would be one way to remedy the problem (assuming there is no yield problem also).
The next best course of action would probably be be to test the yield of the well. You may be able to do this yourself, but a professional will bring in a much larger pump and test the flow right at the well, bypassing all the pressure controls and inside piping that would interfere with the results.
Of course, seeing as you're selling the house, you'll have to make your on judgement call on what you do or disclose -- advice on that is beyond the scope of this site.
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.)
Best Answer
Pump it out for a while. If the pump controller does not have a "low pressure cutoff" pay attention so you don't leave the pump on if you manage to pump it dry.
Common to "shock treat" (chorinate, recirculate, let sit, then flush out) on general principles. If the cap has been in place these 15 years there really isn't anything that should have gotten in, but general principles...
Depending what you are doing (such as drinking the water) might also be advisable to test the water (generally after flushing out quite thorughly and over several days, if you did shock-treat it), to be sure there are no issues with water safety from bacteria or other contaminants.
If your state (or other local authority) has a registration program you may be able to find data about your well from them (as well as, or instead of your well driller) - check to see if there's a plate or something with a registration number for the well on it.