Well, I would suggest to soak it well with thin CA not at pencil end but at their side, do it in steps, let the thin CA penetrate through one side, spray some accelerator after couple of minutes, turn on the other side and repeat until you can see all gaps are well filled with CA.
Drilling can be a problem for unsupported blanks such as those so I would either wrap them if strong tape or cut 4 strips of wood and glue them on all sides like a box, after drilling I would cut the wood strips off put them on the lathe and use the Flap Disc system to get it round close to final size and then finish with sharp chisel or sandpaper.
There are many ways to skin a cat, this is just one of them...!
Good luck,
Cheers
George