John make sure your tailstock MT is clean and your drill or drill chuck is also clean and fit well, (cheap ones can be a problem), then seat your MT well, that is insert it with the flick of the wrist, I like to use a centering drill first to have a good centered start, but you could use a skew point to do that also, when drilling into the material the MT will not likely slip, unless there's a lot of chatter, but withdrawing could possible loosen the MT, so a good prevention is to use a pair of vice-grips to hold the MT (drill or chuck) while withdrawing, something I always did on a metal lathe as well, prevention is better than....
good luck and have fun
Oh one more thing, a drawbar is never used in the tailstock, just wouldn't work, but if you use the drill chuck in the headstock spindle to say hold thin work pieces you want to turn, than you use the drawbar, as there is no pressure to hold the MT in the headstock spindle, it could easily come loose and NO you wouldn't like that to happen,
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