Is the camper plugged into shore power?
The AC does "run on AC" BUT many RV models also require a solid 12 volt DC source to power the thermostat. It is not "manufactured" by the AC since it is available in the camper.
So, by plugging into the truck, that 12 volt DC is available to power the thermostat and the AC runs off shore power as designed.
Now, the converter does sound "bad" (not getting the charging voltage to your battery) so it needs to be trouble shot.
A "condensation induced" short could have damaged the converter portion of your power center, but it could have only blown the fuse in the converter.
You need to disconnect your battery (REQUIRED as a bad battery can prevent the converter from powering up) and then measure the voltage at the battery connection terminals at the power center (with no - or minimal - load). That voltage needs to be above 13 volts.
If not the unit is not working. Check the incoming voltage is 120 VAC (otherwise the breaker is bad) and if good, check the on board glass fuse. If both are "good," you will need to buy a new converter for your power center. Parallax is a good one and you should find an inexpensive direct replacement through internet sources.
If you do have 13 or more volts, the power is not getting to the battery and you need to check IT and the inline fuses between that converter and the battery. (There should be two - one within 18 inches of the battery and one close to the metal umbilical connection box under the frame).
This Progressive Dynamics unit is a direct replacement for yours: http://www.progressivedyn.com/pd4600...placement.html
This website has the kit and the instructions (with photos) so you can decide if it is a project you want to try.
Complete power centers are: http://www.rvplus.com/parallax-power...FQQT7AodK2cASg