74HC595 + ULN2003A not driving stepper motor
Probably just a schematic error, but\$ \overline{\text{ MR}}\$ should go to 5V, not GND, and \$\overline{\text{OE}}\$ (not shown) should go to GND. Also, ULN2003A-COM shouldn't float, it should go to 12V in order to clamp the motors' output voltage spikes to 12 volts plus a diode drop.
Most important, though, is that you're way overtaxing the 2003.
From TI's data sheet, note that the maximum allowed current out of the chip's emitter pin is 2.5 amperes and, ignoring the transistors' Vce(sat), you're trying to put about 3.6 amperes through it [the emitter pin] with all of the 27 ohm coils hot.