my first foul weather/winter commuter was an old mid-90s raleigh hard-tail that I modified with a rigid fork and fenders. it was an awesome machine, but the canti rim brakes and derailleur drive-train were less than ideal on the messiest of winter days. during the winter season i would swap out the slicks on the wheels for studded tires (i had taken a few spills on ice before i got wise to the importance of studded tires).

after that bike was killed by a bus, i upgraded to a Scott SUB 10 hybrid which has an alfine 8 IGH and hydraulic disc brakes. just as on my old raliegh, in winter i swap out the slicks for studded tires. it's a splendid tool for the job, but a little spendy at $1,200, and well outside of your price range. if you're trying to stay on budget, you can do a lot worse than finding an old used hard-tail to upgrade.