The AM stations you are hearing may or may not be overload. Perhaps it is mixing caused by something you did when you added the ground wire. A loose, corroded, or dissimilar metal joint can introduce mixing products in your receiver. The whole topic of external mixing can get really interesting but is too complicated to fully cover here. Either way, whether overload or mixing or maybe something else, it apparently was caused by adding the ground wire because you said reception was fine before adding the ground wire. So, remove the ground wire. My matching transformer has no ground wire attached and the antenna works fine. btw... Be aware lightning protection grounding is another matter.
As far as running on batteries, that is usually best. The batteries provide pure DC whereas many AC adapters can introduce noise.