If you have a leak, then that certainly could explain the the continued pulsing. It does prove both the pump and the pressure switch works which is a good thing.
The lights obviously dimming when the pump runs suggests you may not have a battery connected. This is one reason why having a battery is a good idea to carry the increased current load when the pump runs. As the system uses a pressure switch to control the pump, it's important the pump can develop enough pressure to consistently operate the pressure switch. A power supply that drops its voltage significantly under load may mean the pump cannot reliably produce the pressure needed to trip the switch.
You may also find you need to adjust the pressure switch if the battery is not being charged. The teh pressure switch manufacturers instructions to do this.
Just a few things to remember:-
Drop the pump into the water before you connect it to the caravan. this allows the pump to prime. It's good practice to disconnect the pump from the caravan when you change the water barrel.
Do not allow the pump to run dry, as it needs water passing through it to cool the motor, and to keep the motor shaft seal to activate properly. If you do allow the pump to run dry it will quickly deteriorate and will certainly accelerate its failure.
Always make sure you fully drain the water system when you store the caravan, as any residual water may damage some fittings if it freezes. Check the operating instructions for all the appliances to discover how to drain them.