My home thermostat went bad with the same symptoms recently.
The click you hear(or should hear) is the relay opening or closing. We have a similar coleman thermostat in our 5th W.
I put a digital in out TC about 2 years ago. Much better thermostat than the Coleman. Unfortunately, you already ordered the same thermostat as a replacement. It works.
As far as matching the wires, Old-Biscuit supplied the wiring info. That's a fairly standard color code but rv manufacturers don't necessarily worry about the color.
If it is like my coleman, you may need to write down or take a picture of the wiring. My coleman has the color coded wires coming from the thermostat and attaching to different color wires, with wire nuts, about 6" inside the wall.
