I managed to fix the problem, thanks everyone who took time to provide help/advice. I managed to open the motor but it is impossible to do that without damaging the clips that are holding it in place.
As for the cause of the problem itself: the lubricant that was used on the tiny white gears inside wasn't distributed equally throughout the orange bearings which created this bump that caused the whole mechanism to shift upward whenever the motor started spinning. I removed the gears and applied the excessive lubricant to all of them and closed the motor - which fixed the wobble, but it also fixed that nasty sound it was making.
Hopefully this will help anyone who is suffering the same problem, but beware: you will damage the clips. But it shouldn't be that bad.
