There are two kinds of bicolor leds. One has two leads and one has three. You likely want the ones with three leads (which is to say the same as your common cathode leds) as they are a bit easier to drive in that they are the same as your current leds except they are in the same package and share one lead (which goes to the power rail) each of the other leads gets a resistor and connects to a shift register pin. Setting the pin to 0 (assuming common cathode) turns the led on (red or green depending on the pin) and setting it to 1 turns the led off. Setting both pins to 0 will giver you orange as both the red and green will be on.
Peter