After a long battle, I won. I was able to remove the mast from the motor unit while the motor unit was in the car following these instructions http://www.antennamastsrus.com/Instructions.htm and with the help of vice grips and a hammer. However, the antenna mast sleeve was corroded to the antenna tube, which required that I remove the motor unit from the car in order to remove the sleeve. The sleeve is supposed to come out with the mast, but the corrosion kept it locked in place.

I had to roll the shoulder of the sleeve inward, then pry the wall of the sleeve away from the wall of the antenna tube bit by bit. I used some really small screwdrivers and needlenose pliers. After removing about 1" of the antenna tube, I was able to ram the sleeve down through the antenna tube using the old mast and a hammer. I then used some emory paper to remove the corrosion residue so that the new mast sleeve would slide in easily.

I reinstalled the motor unit, and fed the new mast into place as per the instructions on the website above. Voila! Good as new even if it took a little more elbow grease than I would have liked.