How can I use a single 74595 to drive two different IC's?
You can daisy-chain '595s as much as you want: simply connect the output of one to the input of the next one and connect their control signals together. 

First shift the data for the SN76489 followed by the data for the AY-3-8912. Your code would hardly have to change; just shift 16 bits instead of 8.