The TPIC6A596NE (20-pin DIP) can handle 350 mA. http://www.ti.com/lit/ds/slis094a/slis094a.pdf Unfortunately they are "Open Drain" outputs so they can sink 350 mA but can't source current. I think this means that you can't use them to drive a bipolar stepper where you need to reverse power to the coil (See: Stepper Motors – code, circuits, & construction).
How's your budget? You might need a Dual H-Bridge chip per stepper. At least then you could use cheaper non-driver shift registers.
If you have not gotten the steppers yet, are there any unipolar (5 or 6 pin) options? Those would work with Open Drain or Open Collector outputs.