You have too much voltage drop between the battery and bulbs. Something is wrong in wiring - probably a short. Could be a short on some connection or on some load like faulty bulb or socket.
11.9V at the battery is not fully discharged, in your case. It would've been fully discharged at 11.9 IF there were no loads and IF battery was at rest for several hours.
Turn the loads on - voltage drops, turn the loads off - voltage rebounds to its "true" level after some hours.
Turn the charger on - voltage goes up almost immediately, turn it off and don't turn the loads on - voltage returns to its "true" level after some hours.