I had the same problem with a drill chuck. I eventually came to understand that a drill chuck is primarily intended for use in the tail stock, and when you are using it in the headstock you need to do two things:
1. You should use a drawbar that holds the tapered end of the drill chuck in the morse taper of the headstock.
2. A drill chuck arbor typically consists of two tapers - an MT1 or MT2 that goes into the headstock or tail stock, and a shorter taper (J33)that goes into the drill chuck itself. Even if you use a drawbar to hold the arbor in the morse taper of the headstock, there is still a risk that a lateral force could cause the drill chuck to come off of the shorter taper. The advice I was given was to use the tail stock to apply a slight lateral pressure on the workpiece while turning. You can put a small waste block between the tail stock and the workpiece to prevent digging a hole in the end of the work. (You can turn a small cup that fits over the end of your live center if you want to get fancy.) Or you can use an overlength blank and plan to part off the end as you are nearing completion. You can run the work on drill chuck alone for sanding and finishing provided you keep the speed fairly low and avoid putting any lateral forces on the spindle.