Hi Leda, sorry for the late response.
The water pump should turn on and build pressure (all faucets off and no leaks) and then shut off until pressure drops. It is possible that the one-way check valve is bad in which case it will not hold pressure and keep running. Between that and the low flow, I have to think the pump is bad.
Now for the generator not supplying all the outlets and appliances, there is usually one GFI outlet in the system, probably in the bathroom that also feeds out to many more outlets. If it is tripped all the others will be dead. See if you can find a GFI outlet and reset it. Also, GFI outlets can go bad as well. And yes, yuou can run the generator while travelling.
