David,
I believe patience is the key here. Using a penetrating oil to help free the parts is best. It is likely the dissimilar metals in contact with each other and a lack of oil on the parts from sitting or poor maintenance is causing them to stick. I had mentioned the Leather Pump Seal before and how bad they can get after so long. I have made a replacement out of a piece of leather from an old belt that I shaped over the Spring Guide that the seal sits on. It worked well. You could use rubber "washers" of the correct size or cut to size as an option. After freeing the parts, cleaning them up and before reassembling them do make sure to add a couple drops of oil to the point between the Upper Chuck Assembly where it slides in and out of the Spring Cylinder Assembly to keep them moving freely in the future. A few drops of oil inside the Drive Housing on the moving parts too. A lightweight machine oil like 3 in 1 oil, Zoom Spout Oiler or similar works fine. The same oil to use for the items under TAKING CARE OF YOUR JIG SAW on the top of page 7 in the manual. The Zoom Spout Oiler works well for any hard to reach places.
At the time the Model 10 Shopsmith's and this Jig Saw were made the manufacturers hadn't started recommending using Johnsons Paste Wax as a lubricant and protectant for bare metal parts. Using JPW or similar paste wax has an advantage over oil on parts exposed to saw dust. Saw dust will stick to oil and make a "gummy" mess that has to be cleaned off and redone after about every use. Waxing the outside of the Spring Cylinder Assembly, the Hold Down Support Assembly rod and the openings in the Upper Arm they go into will help keep them moving freely. The PDF below is a supplement to the Model 10E and 10ER Owner's Guides about waxing instead of applying oil to bare metal parts and the same can be applied to the bare metal parts on the Jig Saw.
Even though the oil on the surface will be exposed I still recommend adding a couple drops of oil to the point between the Upper Chuck Assembly where it slides in and out of the Spring Cylinder Assembly to keep it moving freely as opposed to using paste wax there.
Waxing Supplement Model 10s.pdf