Movie : The Blind Side (wiki)
'When Leigh Anne seeks to become Michael's legal guardian, she learns he was separated from his drug-addict mother when he was seven and that no one knows her whereabouts. She is also told that even though he scored low in almost every category in a career aptitude test, he is in the 98th percentile in "protective instincts".
Both are possible, and both are correct. It is partly a matter of style, but also depends on what you want to emphasize. "Drug-addict mother" focuses on her being a drug addict, then adding that she is Michael's mother. "Drug-addicted mother" focuses on her being a mother, but also describing her as addicted to drugs. A writer can choose one or the other.
(I agree with cyberpedant about using hyphens in either.)
In a sentence like that, you can use "drug-addict mother" in the same way as you can say, well, "astronaut aunt" or "championship karaoke singer father." Once you've got his or her or their, we accept a long chain of qualifications, since we know you're talking about one person.
She struggles with the abuse she suffered at the hands of her young-adult novelist grandmother.
He, ironically, finds love at the marriage of his Miami podiatrist-slash-Houdiniesque escape artist second cousin.