Can I increase resistance to cut down on LED power consumption?
Use this LED Series Resistance Calculator to find out your series resistor.
I'm afraid that 6v cannot drive 10 LEDs in series. You would have to try a parallel approach or series+parallel combo approach to drive them.

If the forward voltage is not known guess around 2v to 2.4v and play with the calculator.

You are right, that if you increase resistor value, the LEDs will glow dimmer and your batteries will last longer. When you increase the resistor value, you are reducing the amount of current through the LEDs.