So what’s the difference? Both phrases have a negative connotation and both can be used to say there are (too) many of something.
The most important distinction is that だらけ refers to something is covered in many individual things, whereas ばかり suggests only one from many possibilities.
怪我して体が傷だらけだった。
To have an accident and be covered in bruises.
怪我して体が傷ばかりだった。
(Sounds unnatural—what else could you be covered in after an accident?).