I think you're driving the stepper motor too fast.
With a 1000uSec high and a 1000uSec low, you're producing a 500 Hz square wave. As such, you're trying to step 500 time a second.
Try slowing things down to maybe 50 steps/second (make your low duration 20 mS, or 20,000 uSec).
Okay, turns out that there was an issue with a wiring too and the labeling was indeed incorrect on the motors. A big thanks to Andy for spending his precious time in assisting me.
How I solved the problem
Update: After discussion with one of the commenters below, it is learned that there is no difference in the order of wires connected to the driver, therefore, the order of wires labeled in my question and in the answer below don't matter as much as setting the value on the variable resistor itself.
These drivers have an onboard tiny variable resistor that limits the current to the motor. I just had to tweak it to get everything working. Now the motor runs in ultra low speeds, although with some mild vibrations (in full step mode).
After enabling the micro-step mode provided by the chip, I was able to dampen the vibrations by a huge margin and got everything working butter smooth.
For anyone else who may have this motor, here are some references:
Orange and Brown wires constitute one coil of this motor, with the black being the center tap.
Likewise, Yellow and Red constitute the second coil, with the white wire being the center tap.
The way you connect it to a Polulu DRV8825 is as follows:
I hope this helps anyone else who may have the same issue.
Cheers.
Best Answer
Do you want the good news first, or the bad?
The bad news is that what you are asking for is not available.
The good news is that what you need (rather than what you asked for) is available. On eBay. It's called a "motorized rotary stage". An example would be this unit, which has a data sheet here, although the unit for sale is apparently an older unit which does not exactly correspond to the data sheet. At a guess, though, it will do much better than you need.
And eBay has a lot of this sort of unit. Of course, as in this case they are often old, unsupported, and you'll have to learn how to work with proprietary interfaces.
If you're more adventurous (that is, wealthy), various companies such as Thorlabs and Newport will gladly sell you new units much better than you need. Well, and almost certainly more expensive than you can afford, but you didn't set price limits in your question.