Edam - I walk to the shops too. I have a combination of things. A couple of lightweight nylon bags that fold up really small that I keep permanently in my handbag. A backpack for medium-sized shops. Not a big one, more like a schoolbag type size so I don't look like an actual backpacker which would be embarrassing.
And my ultimate weapon is a fold up shopping trolley for big shops.
Obviously it embarrasses the children but what the hell. On clearance at the moment too, so a bargain.