Keith Newton
Bill, Thanks for reinforcing my suggestion. I started to suggest silicone, except for the fact that it makes it hard to ever come back and get a good finish, if someone ever wants to re-finish the cabinets.

vinny didn't tell us if the doors were stained or painted, to help us make the best choice. If they are painted white or something, I would get a siliconeized paintable caulk. I think the properties would be plenty good for this, and not a problem down the road of time.

As for the outer edges or corners. I think you may be able to fix any leftover problems with flat tooth-picks. Put a little glue on one side, see if you can jamb it into the back space, then lever it over, and break it off, or stab through it with a sharp chisel.