I'm going to echo the crowd, and say don't buy anything new. Your old, cheap MTB is perfect for the mission at hand.
Due to the risk of theft, vandalism, and just plain accidental damage, you don't want anything that is obviously a 'nice bike.'
An alloy framed, belt drive, disc brake hybrid would make a wonderful all-weather commuter, but it'd get stolen inside of a month if it's stored on a public bike rack.

For the weather, use a plastic bag over the seat to protect the vinyl/leather/foam, it's easy to stuff in a pocket while you ride, then tie it over the seat when you leave it for the day.
Use a spray penetrating oil on the chain, and cogs to prevent rust; WD-40, JB80, TKX, etc. It's not the best, but it'll keep the parts moving, and is easy to apply, just give it a quick squirt, and wipe with a rag (or not)
If you can find it, get a spray White Lithium grease, with a needle applicator and get some in to the shifter mechanisms. Again, there's better options, but it will keep the moving parts inside alive when left in the elements.
Get a decent lock; I like one with a cypher instead of a key, so there's nothing to loose, and a long cable to wrap around the frame / wheels, it makes it more akward to steal than a U-lock alone.

I'd even leave the tires alone, at least utill they wear out. With most of your riding under 30 minute trips, going to slicks won't make that much difference, and having knobby tires means that any more-or-less horizontal surface is available to use, if needed.

If you do go with another bike, keep it as low key as possible. I had an older Cannondale MTB (still a $1000 bike) that I had to store outside for a year at ODU. It was successful, other than having a set of $15 BMX pedals stolen off of it, due flat-black paint job, and mostly subdued finishes. (the pedals were anodized purple)

I used the above tips, and tried to rack it in conspicuous places (right by the front door)