If the gauges are correct then you probably have a pressure switch problem or the precharge exceeds the ON setting of the switch.
It is also possible that the pipe to the pressure switch line is plugged and the switch is not responding fast enough. That would explain both the 68 psi shutoff with a 60 psi switch and the delay in turning on the pump when the pressure drops.
Remove the switch and check the pipe and orifice if there is one in the switch. If in doubt, install a new switch.
Following is a possible cause if the switch is known to be working.
The pressure drops slowly as long as there is water in the tank. The water runs out of the tank when the pressure is equal to the precharge pressure.
When the water runs out of the tank the pressure drops instantly to nearly zero and you "run out of water". That causes the pressure switch to operate.
Here is a process that may help to verify and set the air pressure after you make sure the switch is working and if you still have the same symptoms (run out of water before the pump starts).
1. Turn off the power to the pump.
2. Run the tank out of water and shut off the drain when the pressure drops sharply. That is the signal that the tank is out of water.
3. Run the pump for about 10 seconds to get some water in the tank, and shut off the power. The pressure on the pressure gauge (water side) should be the same as the air pressure in the tank.
4. Using a scredriver or other tool, with the cover off the switch, lift the pressure plate away from the base of the switch until it snaps to the open position, as if it reached shutoff. If it won't stay in the OFF position then you DO NOT have TOO MUCH air in the tank. Don't go any farther in this list.
5. Now, with the power still off, slowly vent air from the bladder to let the pressure on the air side drop until the pressure switch snaps to the ON position. This step makes sure there is still water in the tank when the pressure drops to the point where the switch actuates.
6. Turn on the pump and see if the problem is solved.