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。 (Meaning-wise 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 of your four examples below that, it means "only" or "exclusively". In this second meaning, the focus of "only" moves depending on where you put it and which part is accented.
正直に言うと、最近アメリカのドラマを観てばかりいた。
'Being honest, I have been exclusively watching American dramas.'
I.e., "I have not done anything but watching American dramas."
正直に言うと、最近アメリカのドラマばかり(を)観ていた。
Interpretation 1.
'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."
Interpretation 2.
'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."
Interpretation 3.
'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.