Thank you.
Your problem is caused by the pressure switch cutting on and off very rapidly when you spray with a nozzle in place. When you pull the trigger, the pressure drops below the cut on pressure imediately and cuts the pump on. Because of the nozzle restricting the flow, the pressure rises imediately to the cut off pressure and cuts the pump off. When you had the nozzle removed, the pressure couldn't rise above the cut off pressure, therefore the pump kept pumping.
The reason this happens is that water cannot be compressed (only very little) so the cycle between cut on and cut off on the pressure switch is very rapid.
The solution is to provide an air damper to slow down this cycle. This is the same reason that all well water systems have an air tank. You can make one to solve your problem. A simple way is to make a PVC cylinder with 4" to 6" PVC. It needs to be mounted vertically with a cap on the top and the bottom needs an outlet with a 'T' fitting. Connect your hose from the pump to one end of the "t" and connect your hose to the spray wand to the other. The height of the PVC tube will determine the amount of dampening you get. About 2.5' to 3' has worked well for me.
In operation, the air in the damper will compress till the cut off pressure is reached and the pump cuts off. When you spray the pressure still goes down but much slower now allowing you to spray for 0.5 to 1 minute before the cut on pressure is reached. The time is dependent on the nozzle size and the volume of the air in the damper. When the pump cuts on, it will pump till the air is compressed to the cut off pressure. In operation, the air damper will be 1/3 to 1/2 full of the water, so when you finish, you should drain the water from the damper.
Good Luck, I hope this is clear enough.