I have only heard that Mormons get a large number of converts from Alcoholics Anonymous due to the no alcohol or stimulants belief.
When I was LDS, I knew several converts who converted from Catholicism.
However, I did not know any converts who had substance abuse problems prior to converting or were in AA (Glen Beck is the first one I have heard of). I did know several people who grew up LDS who were addicted to prescription drugs, used illicit drugs and alcohol in High school, and 2 who drank for ‘medicinal reasons’.