it is possible the fuse between the converter and the battery is blown.
That still would not explain your brown outs completely. When your converter is on, it bypasses the battery
. A blown fuse between your charger and battery
would certainly explain why it wasn't recharging tho.
As mentioned above, with the converter on, you can pull the battery out of your rig and STILL have 12v power. Thats what a converter does. Many folks here do not have a battery, they run strictly off the converter because they only camp with shore power.
I don't know about other models, but mine throws a relay which switches on the charger, and then disconnects the battery from the 12v system while it is charging from the charger built into the converter. The converter, and *only* the converter supplies the 12v system for the house.
I have breakers on the converter unit which also serves as a 110v distribution system. I also have a separate breaker to turn the converter/charger section off. It is set permanently to "Off". The charger that comes with the converter stinks, so I use a small outboard one instead. When I am plugged into shore power, the charger charges the battery and I run off of it.
I think the "Brown Out" theory in the park is a good one, or you are somehow not recharging your battery.. but why your system would be running off the battery when the converter is on, I do not know. Once you switch the converter on, it should drop the battery out of the house feed.