I want to cut wooden disks from a wood board. I bought what I thought was the right bit for my drill. However, the bit seems to be solely for the hole, not for the disk, as it comes with a centering drill. The drill will drill a hole in my disks, making them useless for my purpose. It tried to tighten the centering drill further in, but that did not work. The tightening screw seems to only work on the flattened part of the drill and (not surprisingly, when you think about it) on the thread part of the drill. When I push the drilling machine down, the cutting part will push up along the drill.
How do I solve this? Is there a spare part I'm likely to get at a DIY store? Do I have any chance of re-purposing some other drill bit for this?
Edit: I have a stand for my drill, so I don't think I actually need the centering drill.