>I'm having a hard time finding where it's suggested in their literature that salvation comes from sobriety?

The message is the opposite. It's sobriety comes through salvation.

1 We admitted we were powerless over alcohol - that our lives had become unmanageable.

2 Came to believe that a Power greater than ourselves could restore us to sanity.

3 Made a decision to turn our will and our lives over to the care of God as we understood Him.