You may well just be looking at poor connections from your battery to your converter. A weak connection will act like a huge load to your battery and not even be something you notice.
Wiggle the positive and negative wires in the converter terminal strip, tighten everything you can.
I had a very loose negative wire at my converter, after 4 days plugged in the battery read 11.9 volts and everything was shutting down. I wiggled the wires, and bingo, lights suddenly came bright inside the rig as converter power got to the battery.