Coil-less AC-DC line regulator
Moving my comments to the answer space so that others can get use out of it:

Have you tried putting the cap on the reference pin (2) of the shunt regulator? You want the reference voltage to stop oscillating which is why I would think it would do more there. If you put it at the cathode, it's like filling up a cup with a fire-hose and expecting that you'll be able to accurately get a certain amount of water in that cup.