A day-pack size knapsack with a padded hip strap works out well for me; it's stable and comfortable even when loaded with 20lbs of groceries. The weight is carried higher than is considered ideal for biking, but the messenger bags look like an awful idea to me, and I had the added constraint that I want to go on outings for which I'll lock the bike and take the stuff with me, for which bags affixed to the bike look too clumsy.

The only time I've gotten things soggy in it is when I've taken off the bag and placed it on a wet surface; I do keep small things in ziplocks, and add a grocery bag around my laptop if it's actually raining. (My clothing's a different story; I either wear fleece and drip-dry when I get there, or I carry a change of clothing.)