+1 on the hybrid, or a rigid MTB with slicks/simi-slick tires. Especially for your short ride.

Keep an eye on Craig's List. It's estimated that three quarters of hybrids don't get 100 miles on them until they're stuck in a corner of the garage. Eventually they're sold for a few hundred dollars to clean out the garage.

You could get a bike with lights, a rack and bags, new tires and tubes and flat repair stuff along with a Showers Pass jacket for you. Set to go to work with everything within you're budget for the bike alone.