There are two different issues being discussed here.

Dogs behaving normally will eat as much food as they have access to.
(Please do not reply to this statement! Different breeds and individual dogs make their own rules - I know!)

The other issue began with non food items being eaten and chewed.

The last thing you want to do is exchange a treat for any item the dog has in the mouth that it shouldn't have! Every time you do that you are telling the dog that it will be rewarded for that behaviour - you are training the dog to do it. The dog will understand that every time it brings you an item it gets a treat - that is a good thing as far as the dog is concerned. The dog will then go look for another item to get another treat....

The fact that she only does it while you are around suggests that the dog is in charge of you. If Nana ignores the behaviour (for whatever reason) the dog gets no attention or other reward and so the behaviour is a waste of energy. Take a leaf out of Nana's book!