Energy stored in the battery, measured as mAh, has nothing to do with the batteries rated supply current, measured in A.

7 of your Futaba 3003 servos will drain no less than about 1.5A just by turning them on. You can expect peak current closer to 7A. If your battery (please link it here) can only supply 5A peak, than your servos will suffer in torque to compensate.
Also, as Joker94 said, 1000mAh is too low. If you're draining 3A on average, that means your battery will be dead in about 20 minutes (1000mAh/3A = 20 minutes).