Well, glue doesn't dry in the bottle, so no reason why it would dry
deep inside the seam.

You mention a paper separator - why? Just glue the blocks directly
on, and cut them off with the gouge later.

Also, try a two-part epoxy. It doesn't require water or air to cure.

When I use yellow glue on the lathe, what I end up doing is this: I
rub on an even coat of glue using my fingertip. Then I use the
tailstock to compress the two parts slightly, and turn one of them
(i.e. spinning the glue). I lock the headstock for this. As I turn
the wood around, I increase pressure on the tailstock. Between the
pressure and the motion, I get most of the excess glue out, and leave
a very consistent glue thickness behind. When the glue seems to
"stick" and it gets hard to turn the wood any more, I move it to its
final orientation (I'm usually gluing up segmented rings) and tighten
the tailstock a bit more. I can true rings after an hour of drying,
but for a bowl blank I'd remove it after 10 min or so and let it dry
the rest of the way overnight.