The "master switch" is likely a relay, so the problem may be either the little switch that triggers the realy or the big relay located elsewhere, usually near the batteries. But if your "master" is a large switch, usually one that rotates rather than toggles or rocks, then it is probably a direct switch.
When plugged into shore power, you should see something like 13.3-13.7 volts, not just 12.0, so I'm wondering if the problem may be something else? A fully charged battery reads 12.6 volts and an active converter/chager (on shore power) should put out 13+ volts in normal operation.
What type of batteries do you have and what is their age? Assuming they are not sealed batteries, have you checked the water level to see if each cell is full?
2004 American Tradition; 2014 Buick LaCRosse
Homebase in the Ocala Nat'l Forest near Ocala, FL