And of course if all else fails, you could just reset your Internet Options to "never dial", like I do. That way nothing, not even IE, can dial out, but will simply give you a "page not accessible" error if you haven't already dialed up yourself.
If you go that route, check that setting from time to time too. You may find sites and/or programs that'll try to change it, similarly to how browser hijackers work. And those fall into the rather fuzzy borderline area between worms, which avast should catch, and spyware which avast can't see but a good utility for the latter should find.