If you don't want to carry around dress shoes then just buy some overshoes. No need to complicate things. If you're worried about how dumb they look, don't. The secret is that everyone wonders what shoes are so fancy that they have to be protected in such a way. For those occasions when you forgot your galoshes, be sure to keep a couple plastic grocery bags handy :)
Good idea with the over-shoes....SECONDED.