In your price range and riding conditions I'd look for an 80's mountain bike with rigid front fork(No suspension). For around $150-200 you should be able to find a quality brand bike in ready to ride condition. The market is weak on old MTB's so you can really get a nice bike with good quality components. Another nice thing about the 80's mtb is they usually have mounting options for fenders and racks so you can use it for almost anything. They also don't attract as much attention from thieves.
I have been using an old beach cruiser with balloon tires to get back and forth to work the last two weeks , have only slid out once and that was on black ice - other than that it has handled the slush and ice pretty damn well