Yes you could use that darlington array to bypass all the clutter (however a 15 pack of transistors is only like.. $4.50 at radio shack), but would definitely help in the wiring process.
Now you can sort of... bypass the shift registers, but you'll still end up having a wiring mess on your hands cause you will effective be going from 6 outputs on an arduino to well... you don't have enough outputs on your arduino to accommodate all the wires..
Wait scratch that. YES! Actually using one of those darlington arrays, and TWO transistors, you can cut the clutter in 1/2.
Wire all inputs (well negatives from the displays) together so you only have 1 wire per input for both leds. Now for those inputs, wire each of the 8 inputs to the darlington array.
And you will need two transistors to switch the VCC voltage for each display. This way you just constantly write to each of them at one time. You shouldn't really notice any flicker with this setup. More than two displays however, it could get ugly.