The drill press is a fantastic machine for accurate and repeatable boring of large and small holes, but adjusting the table for drilling angled holes is always a chore, and resetting it to 90 degrees is almost as difficult.

Woodpeckers Duax Angle Drilling Table allows you accurately drill angled holes at the drill press without having to adjust the tilt of the machine's main table. The table can be tilted all the way to 90 degrees for drilling into the end of long material. The toothed stop can be used to lock the table in at any whole degree increment and the dual lock knobs and robust pivot hold the table at exactly that angle.

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Start with a smaller bit (3/4", e.g.) and increase the size progressively (1-1/4" then 2", etc.). And use dust extraction.

The height of the jig and the hollow core would make the vibration worse.

@JINRO Also, are the teeth of the drill bit engaging with the wood evenly and consistently all around the rim? If only half of the rim is engaging, the forces are uneven and the bit will want to push the material around. You could try adjusting the RPM of the drill press or feed rate, as well.