I think I would go for a touring bike, in your situation. Not as light as a road bike, but it will have the attachment points for the fenders and rack, have a more comfortable riding position, and will also have lower gearing to make the hills easier and less sweaty to climb. Just take your time and climb up to work at 5 mph or less, if you have to, to stay dry. Just need the gears to make that happen.

Also, consider a recumbent. They're more comfortable, and offer more torque for easy climbing.