I would start by checking the pressure gauges.
Pressure dropping to 15 and then jumping to 50 psi when the pump starts sounds like too much air on the air side of the bladder.
Check the air-side presssure when the pump shuts off. It should be exactly the same as the water-side gauge. If it is not then the gauges are not matched.
With the power to the switch off, slowly discharge all water from the tank while watching the water gauge. The pressure switch should actuate to try to start the pump 2-3 psi before the tank is empty.
The pressure on the water gauge just before it drops quickly is the precharge air pressure in the tank.
Check the air pressure again when the switch acutates to start the pump at the low end. It should be the same as the water gauge pressure.
After you get the air pressure and the tank pressure matched on your measuring devices, and are sure that the gauges are giving you the correct readings; try the cycle again and see if you have solved the problem.