I had pretty much the same thing happen a couple years ago. I had kept the RV plugged in and the battery connected while at home, and eventually the battery reached the stage that it showed full charge on the RV's built-in tester, full 13 volts on a volt meter, but it was incapable of actually delivering any current. Throw on the light switch, the light came on for a split second and immediately went out.
The solution was twofold: (1) took the battery to an Autozone, and they tested it and confirmed it was in that condition, so I bought a new battery; (2) I no longer leave the RV plugged in with the battery connected. I now disconnect the battery and hook it up to a BatteryMinder when I'm not using it. If I need to go into the RV and need lights for a bit, I'll either reconnect the battery for a bit or plug it in and unplug it when I'm done in there.
