The socket for the AC adapter usually incorporates a set of contacts to switch the radio from battery to AC. Sometimes they corrode. Try plugging the AC adapter plug in and out and wiggling it a little to see if it will start working off the batteries. Do it with the radio on, the ac adaapter NOT plugged in and the batteries in, so you can hear if it starts to work.
There must be some kind of damage on the inside that prevents it from using its batteries. I have a friend who's an electronics repairman. Maybe he can fix it. If not . . . I'll just have to "live with it" (as a fellow FReeper put it). But what a pain in the tail!