What's happening is that you're drawing more current from the battery than it can comfortably produce. The voltage is dropping due to the excessive current draw, and causing the cutoff to activate.

Adding a 7th cell would only make the problem worse. Increasing voltage also increases the amount of current drawn by the motors.

The obvious solution is to get a battery that can handle the current. However, you may be using a propeller that is too large on these motors. What size motors and what size propellers are you using?