[Ron Ramirez]

Ah! I remember now...each screw is held captive within a dowel that is glued into the door where your doors now have empty holes.

In this case, it does not matter what size the screws are as long as each one can be made to fit within a 3/8" wooden dowel.

You'll likely need a drill press for this. You will want the screw hole to be centered in the dowel.

Yep...this will actually make it easier for you to replace the knobs.