Can you leave some of the clothes at work? That will make things easier. In any case, any waterproof pannier can do the trick and ortlieb is a pretty standard answer.

Cheapest solution would be a small duffel bag that you strap on to the rack (which you already own). I'm indifferent to whether a bag is waterproof or not since I put my clothes in plastic bags before putting them in a pannier.

Jandd makes a very good duffel rack pack that I have used a lot for the purposes you describe. It is easy on and off the bike; it is very stable and, like all Jandd products, very nicely made. With a shoulder strap, it is pretty easy to walk around with the bag and it doesn't have any hooks to dig into you:

Duffel Rack Pack