Yes, I think it does. What begins as a "psychological game" is the first step towards a mental adjustment to a new self-identity as a nondrinker. "Can't" implies an unwillingness or inability -- whereas "don't" implies volition and determination. Big difference.
Others here are better versed on AVRT than I am, but once you recognize that ANY thinking that leads to taking a drink IS the addictive voice, you are on your way to learning to identify it. The addictive voice will tell you that you "can't" drink, for example, implying it is against your will, and you really do want to drink.
Self-identifying yourself as a nondrinker makes all other decisions very simple. You simply do not drink. Period. End of story. No explanations or reasons required.