A pressure tank should have enough drawdown to let the pump run for a minute or more. The drawdown of a pump is on the order of 25% of the volume. So if you had a 7 GPM pump, your pressure tank should be about 28 gallons or more. That is an oversimplification, but it is roughly right. So that 7 GPM rate would cause you to take about a minute to fill a 28 GPM tank to where the pressure switch would shut off. And that presumes the pressure switch is right at the pressure tank.
A 2.8 gallon tank would fill in 6 seconds, 1.4 gallon tank in 3 seconds and so forth.
Now if the pressure switch is before the pressure tank, the pressure from the pump will shut off the pump sooner, because there will be a pressure drop between the switch and the tank. If there is a 2 PSI pressure drop, that 3 seconds would become one second. Then when the pump turned off and the pressure equalized and the water is maybe also being used by the shower, the pressure switch may turn on the pump very soon. So you get the really short cycling.
Some pumps are 10 or 20 gallons or more.
You need a much bigger pressure tank, or a Cycle Stop valve, or a Pside-kick kit (which includes a CSV and a 4.x gallon pressure tank). An advantage to that kit is that it places the pressure tank at the pressure switch. Or you could use a CSV before your pressure switch and pressure tank. That slows the water flow to 1 GPM after the set pressure and the small tank fills more slowly.
If you still have access to the failed 83 gallon pressure tank, that could work by adding air periodically. I don't know what period that would be (week?) and there is no easy technique that I know of to get the right amount of air. Maybe you can identify the air-water level by tapping the tank.
But if that is gone, you need a bigger pressure tank or CSV. And your pressure switch should be at the pressure tank. But that pressure switch location is only part of the problem.
