I think you're starting with a faulty assumption--i.e., that declining a drink will automatically result in people thinking you've got a drinking problem. That's not necessarily the case. When people ask me if I want a beer, I just tell them, "nope, I'm good." And that's the end of it. If someone does press you about it, just tell them you're trying to live a healthier life. That is both true, and doesn't necessarily imply that you're not drinking because you're an alcoholic.
In short, as you already observed, I think you are overthinking it.