Loose Veneer, best Fix?

Once you find the right cement, and procure a set of clamps (Which are an absolute MUST!!!) get a sheet of wax paper. When the glue is in place, and before the clamps are tightened down put the wax paper where the glue has been placed to prevent the clamps from being glued to the cabinet. It would also be advisable to put something between the clamps and the wood cabinet so the tight clamps don't damage the wood.

In tight spots, wooden toothpicks make nice glue applicators. They're also useful for repairing stripped screw holes.

Elmers glue is fantastic stuff, but with the old adhesive residue it will not bond well. Try to use an adhesive intended for veneering.