Well pump short cycle diagnosis
Sounds Like: Waterlogged pressure tank, as per usual. No (or little) air, no (or little) pressure storage, since water does not compress.

With "modern" bladder-type tanks, that's usually a bladder failure, and usually time for a new tank. (In my personal experience, a tank with a replaceable bladder sounds nice, but does not pan out in practice: price of new bladder, no warranty, almost equal to new tank, 5 year warranty. The whole tank is easier to change out than the bladder in many cases.)

You can make a temporary kludge on most tanks by draining the tank, pumping it up (probably to 37 PSI assuming a normal 60/40 pressure switch - in any case 2-3 PSI below the "cut-in" or pump start pressure) and then turning the pump back on. This will buy you some time until that air bubble dissolves (what the bladders prevent, when intact) to get a new tank - or, rarely, very rarely, to find that this isn't the problem at all.

"Why the wild pressure swings?" With no air to act as a "spring" the pump pumps up to 60 and shuts off, but using a very small amount of water drops the pressure and the pump kicks on, then shuts off, then kicks on - this is, by the way, bad for the life of the pump, which is one reason to get this fixed as soon as possible. With the air acting as a spring, several gallons of water are expelled between 60 and (typically) 40 PSI, then several gallons of water are added as the pump runs, then several gallons of water are expelled again. With a seriously waterlogged tank, it might be more like a 1/4 gallon between 60 and 40 PSI, and it drops to 10 before the pump manages to get started.