The motor voltage rating is just a nominal figure, and basically tells you the minimum voltage you need to get maximum torque out of it.
The 0.38A stated looks like the maximum RMS current (so 0.54A peak) the motor can continuously operate at, without burning up.
It looks like an unipolar motor (6 wires), similar to
this one. These are often run from Darlington arrays (something like ULN2002A), not via H-bridges or proper stepper controllers.
If the black and white wires (or any pair of the six wires) are shorted, you cannot use the DM556 controller, because then the stepper cannot be used in bipolar (4-wire) configuration. If they are not shorted, then you can use stepper in bipolar mode, and the DM556 controller in software configured current mode (SW1:OFF, SW2:OFF, SW3:OFF), with the controller set (using the Windows ProTuner software) to use 0.5A, and SW4:OFF (so standstill current is halved), and preferably microstepping (4 or more microsteps). Even then the motor may overheat eventually.