The short answer is no. As to the long answer...
I think we can all agree that alcohol would not fall under the categories of "dangerous drug" or "controlled substance". That leaves "drugs" as the only possibility. While there are compelling reasons to consider alcohol a drug, I do not believe it applies here.
First, the Texas statutes dealing with drugs do not list alcohol. Additionally, it would have been quite easy for the legislature to add the word "alcohol" to the statute if they wanted it there. This simple addition would have made the statute clear and unambiguous on its face, as the legislature desires. Since it isn't there, the only logical conclusion is that they did NOT intend alcohol to be considered.
Also, assuming you read the entire list, you will surely agree that the listed charges are more serious than DUI; inducing gang membership, tampering with consumer products, leaving a child in a vehicle, stalking, dog fighting, compelling prostitution, to name a few.
And, even if I were incorrect, one could still argue that the "may deny" a license language still permits discretion. With the charge being relatively non-serious, just a misdemeanor, and so old, 8 to 10 years ago, I simply can't see you being denied over this.