Get a hypodermic needle or restraunt ketchup bottle and put wood clue in it. squirt the wood clue under the veneer and use clamps to press the veneer down. If it is really bad you might want to take your cabinets to a antique furniture restorer.
We just did all our cabinets on our 1960
and they look great. Save any broken bits that fall off. Glue, clamps and save any sawdust from sanding. If you have pits you can take the sawdust and mix it with wood clue and fill in holes.
I know from other furniture I have had restored the antique restore shop has a product that will take the place of any missing veneer. They did a great job.