You could try searching for... well, wait. There's two things.
Dual Diagnosis: Where someone has a diagnosable mental health disorder like depression in addition to being an alcoholic/addict.
Cross-Addiction: Where the alcoholic/addict has another compulsive type disorder like gambling, sex, food, shopping, etc.
Depending on what you're looking for, if you search for either dual diagnosis or cross-addiction (with or without the hyphen), you should find plenty on what you're looking for.