Shut off the breaker for the CONverter. If they are powering everything, maybe the converter is trying to charge the battery. Are you monitoring your battery voltage? Your INverter is going to hammer 2 6 volts. You will need many hours of charging if your converter is only charging at 13.6 or so volts.
At minimum get a digital voltmeter. If you want to go with a good monitor that will let you know more about how many amp hours you have used, etc, get something like a Trimetric. Your battery should be 12 volts at the minimum. Everybody has different likes or methods but you basically want to go no lower than 12 volts.