try reading up on stepper motors, as the unipolar driver should work fine if you have a unipolar motor of similar size.
You won't get much going or learn much just by following assembly instructions. Electronics is a methodical process: you generally need some understanding of the underlying theory in order to troubleshoot. Read a few articles on steppers, the ULN2003 datasheet, and research any terms you don't understand.
So step one is to figure out why your ULN2003 burned up! Then if you get it working, steps 2, 3 and 4 will come naturally.