My drill motor also has a locking switch. I turn my cork on the blank. Never had any issues with the blank splitting. A little thread wrapping at the end helps.
I tape up an old epoxy brush handle so it fits snugly in the blank end and chuck that up. I also use a sewing machine foot pedal for hands free motor control. I made my own rollers for the lathe and epoxy drying.
One thing to watch for is too much blank past the last roller. That can cause the blank to whip around while turning.