Fair Oaks is still a bit south from here, I'm north of Napa County and East of Mendocino County. If you were close by I would stop by and take a look at your system.
The duty of the converter is to take the 110 volts that you hook up to at the campsite or your home and reduce it down to 12 volts used to charge your battery and run your lights. When you are hooked up to 110 volts using a converter it will not draw the 12 volts from the battery but use the converted or reduced power to run the lights and other 12 volt systems. You only use the battery power when you don't have the capability to hook up to 110.

If you don't have a converter built into your teardrop then the only way you are going to be able to charge your battery is by either using a battery charger (which is a converter) you can pick up at any automotive supply shop or RV center or by hooking it up to your tow vehicle and charge it when you are traveling. It is important to either use an isolator, (an electronic switch that will not allow you to drain from your tow vehicle battery if it's hooked up) or disconnect it from your tow vehicle when camping and using the 12 volts.
If you have a 110 volt plug in (which it looks like you do from the photos) and it doesn't go to a converter, (changing the 110 volts to 12 volts) then you must have a 110 volt outlet somewhere in your vehicle that will be hot only when you are plugged in to 110 volts. This will be true whether you have a converter or not. Regardless, you should have a least a circuit breaker somewhere between where you plug your rig into the 110 volt current and where you can plug in a 110 volt appliance. If you do not have a converter then the 110 volt circuit is isolated from the 12 volt circuit which is usually used for running the lights, your fan or charging your cell phone through a outlet that looks like the cigarette lighter in your car.
Hope this makes since.
