RE: Troubleshooting the fresh water pump
The key item in what you said is that it worked intermittently but better
with the engine running.  That hints at voltage drop in the lines.  Check
all the connections especially those spade connectors on the top of the
water tank.  Then check the voltage coming to the tp of the tank from the
switch on the faucet.
So, in order most likely culprit:  Connections at the top of the tank,
switch at the faucet and it's connections, then the pump itself.