A good messenger bag like a Chrome work great for commuting or around town but I still perfer to use a pannier for heavy loads like groceries. I use a single Arkel Utility Basket and my Chrome for groceries. I put all of the heavy items in the Pannier and carry the remainder in my Chrome. A single pannier does not case any handling problems so if you don't need the space 2 panniers are not necessary.
Still the suggestions to get a different errand bike I not a bad idea. I have two different bikes one setup light for good weather commuting and fun rides and the other setup with racks and fenders for errands and poor weather. You don't have to spend alot of money on the errand bike. As you probably won't be riding it alot of miles.
Craig