*sigh* I ride in the rain much more than I would like to....
I've found that there is little I can do to really keep my feet perfectly dry. It's usually also cold when its raining so I've been wearing storm socks (its a brand name of water proof socks - they are great!) wool socks and then Sugoi booties over my shoes. If it really storms my feet still get wet - a lot of it comes up through the bottom of the shoes around the cleats. I've put silicone caulk around the cleats to seal it up as good as possible and it works when its not really dumping. I think water probably also travels down my tights and into my shoes when its really, really wet out.
As far as drying my shoes goes, when I get home I take out the insoles and put the shoes over one of the furnace vents (we have a high efficency furnace so the air never gets really, really hot) They are usually dry by the next morining.