There is no such thing as “properly cycling”. If the pump is cycling on and off while the sprinklers are running, that is your problem. The amp spike staying high after start up is because it has been starting too many times. This will cause the overload in the motor to trip, which lets the tank empty and causes water hammer. In a couple of minutes the motor will cool off and the overload resets itself, and it will work again for a while.
That 1 HP pump will probably produce about 10 GPM. Your well only recovers at 4 GPM. So you either need to keep running 4 GPM to the sprinklers and let the pump cycle itself to death, or install a Cycle Stop Valve (CSV). The CSV will turn that pump into a 4 GPM pump, so it won’t cycle and trip the overload.