I'm sort of surprised no one has suggested a fixed gear road/cross bike with wider tires. I commute year-round in the Boston area, and I use a Surly CrossCheck built up as a fixed gear, with full fenders and a rack. I really do think that, once you're used to it (and it doesn't take that long), a fixed gear gives you better control in ice and snow, all else being equal (yes, I use a front brake and you should too). And you can keep road geometry and a fit that's more like what you're used to. I use ~32mm tires generally, not necessarily knobby 'cross tires, but the heavier, cheaper, flat-resistant city variety. You don't really need studs or knobbies, unless you're spending your whole commute on bike paths or something.
Older road bikes to convert are easy to find within your budget, and many have enough tire clearance for 28's or 32's, even with fenders (and you'll be a lot more comfortable if you get fenders). Or alternatively, there are a lot of cheap road/city fixed gear bikes on the market now. If you're going to buy something new under $500 one of those is still a better deal than a cheap MTB that has a lot more parts and therefore has to hit the same price point by making even more compromises in quality.