I have a 6-drawer dresser. While moving it, I learned the hard way that
the top three drawers were secured to the bottom three by a long, 1/2
inch long x 1/2 inch wide mortise/tenon joint with three wood screws
driven up into the tenon. One such arrangement on each side of the
dresser. The top three drawers came off, pulling the screws out of
their holes.

Here's a link to a picture of the problem; the bare wood in the middle
is the tenon:

http://www.dahdistributing.com/joint.jpg

I cannot remove the screws because there's wood glued and nailed over
their heads. It's not coming off.

I CAN get the joint back together and the top tier rests solidly on the
bottom tier. The tenon doesn't shift side to side in the mortise. The
screws just slide up into what's left of their holes.

The wood is cherry, I think. This dresser is older than I am (50+
years).

What glue would you recommend?