My lathe tailstock is misaligned

Given the fact that lathes can be used to drill very precise concentric holes, I thought a fun project would be to make a fire piston.  So I took my default 1/4" brass rod stock, chucked a 3/16" bit into the tailstock and started drilling....

... but to my surprise and disappointment, the hole burst out the side!

Clearly the tailstock isn't quite parallel to the lathe axis.  Upon further inspection, it doesn't look like the sole plate of the tailstock is worn badly, and the spindle fits snugly.  However, the barrel has a bit of vertical wobble, and appears to droop down a little bit.  I guess the barrel has worn the bottom surface of the barrel ways in the upper casting a very little amount -- maybe a few hundredths of an inch.

I guess if I need to drill a deep hole, I'll need to do something like the suggestion in this article, which involves modifying the tailstock so that its vertical axis can be tweaked.  Sadly, it looks like I'll have to do something different than is shown in the article, because my tailstock has a horizontal cross slide (note the adjustment screw).