It actually has two meanings. When it is put after the predicate like your original example, it means "just now", and it conflicts with
最近 "recent". That is why your sentence was corrected.
# 正直に言うと、最近アメリカのドラマを観ていたばかり。 (Meaningly odd)
'Being honest, I have just now been watching American dramas recently.'
If you put it within the predicate phrase like the first three sentences fo your four examples below that, it means "only" or "exclusively". In this second meaning, you have different focus of "only" depending on where you put it and which part is accented.
'Being honest, I have been exclusively watching American dramas.'
I.e., "I have not done anything but watching American dramas."
'Being honest, I have been watching exclusively AMERICAN DRAMAS."
I.e., "(I may have done activities other than watching something, but) I have not watched anything other than American dramas."
'Being honest, I have been watching exclusively American DRAMAS."
I.e., "(I may have done activities other than watching some American things, but) I have not watched any American things other than dramas."
'Being honest, I have been watching exclusively AMERICAN dramas."
I.e., "(I may have done activities other than watching dramas, but) I have not watched any dramas other than American ones."
I have no idea why you have the last sentence. It is almost the same as your original sentence, and does not improve anything.