I recently polished my Corian, but it sounds like your scratches are deeper than mine.
I used 3M Imperial Hand Glaze polishing compound (found at auto body supply stores) with a high speed buffing wheel. I had to make 3 passes to get the scratches out. After that, I sealed the Corian with a good polymer automotive polish and it looks great. Liquids now bead up on the surface.
2005 Fleetwood Providence 39J
CAT C7 350, MP-8 Power Module
My wife does all the driving - I just hold the wheel...