I have not done refinishing, but can pass along a tip. I am on my second set of Corian countertops. The first set was polished to a high gloss, and was hard to keep looking good. I had it professionally polished by the installation folks every couple of years.
The second set is polished to only a low satin type of gloss. It looks better with the dark color and is much easier to keep looking nice. Between the lower gloss and the darker color it looks great after 18 months of heavy use.
So maybe the answer is to take it all to lower gloss level and leave it there. You could experiment on a scrap if you have any.