Bottle stopper problems

OK, they have to be one of the simplest of turning projects, but I’m having a problem. I bought a pack of corks – the ones that are meant to slide over a dowel. I have a Jackob’s chuck, so I drilled the blanks, glued in the dowels and let everything dry over night. I chuck the blank in and rough it to round with the tailstock in place – in fact, I turn as much as I can that way. However, when I remove the tailstock to turn the very top, the dowels break almost every time. My gouges are freshly sharpened, and I try to do very light cuts, but 8 out of 10 times, the dowel breaks just in front of the chuck. I’ve tried dowels made of poplar and oak – same results.

Any ideas as to what I’m doing wrong?

Thanks!

-- To do is to be