If you turn on shore power, what happens? Does everything work but the water pump?
With shore power on, the converter is supplying the 12 VDC and the batteries are basically just redundancy. So for now skip checking the specific gravity and all that and reconnect the shore line. Does the pump have 12 VDC? If it does then the pump is most likely bad. If there is no power to the pump, then you need to determine where the fuse or circuit breaker is and check for power on both sides of the fuse or CB.
If this doesn't make sense to you, I would suggest a mobile tech. If you're in a campground, ask for a recommendation from the desk clerk. If you are at home check Google or the Yellow Pages for one.
Extreme heat or cold can cause strange things to happen to electrical devices. Extreme cold can freeze water, could your pump be frozen? Is the pressure sensing switch on the motor bad? There are a lot of variables and that is where experience comes into play. If you don't have it call the moble tech.