Remove the doors from the cabinets, using your screwdriver to take off the hinges. Set them on a work bench.
Sand down the surface of the doors with your sander. Don't try to sand off the existing veneer; just get it dull and flat, removing the shine.
Measure the door in both directions. Transfer the measurement to a piece of veneer. Add an inch all around. Mark the cut with your pencil and square on the back of the veneer, over the paper backing.
Cut out the veneer with your utility knife, cutting through the paper backing and deeply scoring the veneer. Snap off the cuts.
Pull the paper backing off the veneer and immediately press it to the door, leaving an inch all around. Press it firmly with the heel of your palms at various points.
Wrap a wood block in a dishtowel and run it firmly over the surface of the veneer to completely secure it to the door. Trim off the overhanging edges with your utility knife, cutting alongside the edges of the door to get it straight.
Repeat for each door. Let the veneer set for a day, then re-assemble the doors.