Do you need to individually control 6 bi-color LEDs? If you don't have 12 pins to spare you could always use more shift registers.
Did you want to control all 6 bi-color LEDs together with two pins? I think that can be done with two transistors each:
+5 ----- Resistor ---(a)--- LED ---(b)--- Resistor ----- +5
Use NPN transistors to ground point (a) for one color and point (b) for the other color. The transistors should be able to share a common data pin but may need separate base resistors.