Heartland Cyclone Water Heater

On my water heater, the ON/OFF switch on the heater itself keeps melting. Right now we're running propane only.

Are you trying to operate it on gas or 120V? Because if you are fulltiming and plugged in you should be operating on 120V and the wires to check for voltage are black and white (not brown). If you are trying to operate on gas then the trouble shooting is a whole different ballgame (I.e.: good battery, good converter, 120V power, fuses at converter, gas source, etc.)

Unfortunately we really need a model number on that heater if possible. For instance in the 6 gallon models there is a wire color difference between a G6A-3E and a G6A-7E.

Aside from that I would use a small flat blade screwdriver and pop the switch out of the wall panel. Check that 12 volts is getting to the switch and if it's getting through the switch to the downstream side. 

Based on your picture with a brown wire at each end of the board it looks like:

When on Electric you should see 12 v between the white wire (+) and the green wire (-) at the plug.
When on Gas you should see 12 v between the orange wire (+) and the green wire (-) at the plug

Of course I'm guessing at the model so the wire colors might be different.

Here's a link to the manual I'm looking at.
Link