Oh, and if the problem was with the adaptor, there are no user-serviceable parts inside there.  You have to replace the whole adaptor with a new one (which dell will do if you are under warranty).

Replace the adaptor with a different one and see if the problem goes away.  If so, the previous adaptor was faulty.  If not, try a different battery.  If that solves it, the previous battery may have been defective.  If neither improves the situation, there is something wrong with the laptop itself.