Sounds like the relay on outlet A is faulty. Solid-state relays can be damaged by overload or power surges to a point where they no longer switch off fully, and I would guess that is the problem with yours. We use electro-mechanical relays which, perhaps surprisingly, we find to be much more reliable.
Ok, I wonder how difficult that is for me to repair? Since outlet B still works, all is not lost, but still the situation is not ideal.