If you were under 21 and it was your only alcohol-related offense prior to 21, you can get an expunction from the JP regardless of the disposition of the case. If you were 21 or over, you'd have to go through district court and would only qualify for expunction if the case was dismissed outright, if you were acquitted, or if you successfully completed deferred disposition. If you were 21 or over at the time of the offense, and.you were convicted, there is no way to expunge the records.
If you received one, and only one, conviction for an offense under the Texas Alcoholic Beverage Code (TABC), while under 21 years of age, it's possible to get that conviction expunged. "Any person convicted of not more than one violation of this code while a minor, on attaining the age of 21 years, may apply to the court in which he was convicted to have the conviction expunged. Section 106.12 TABC. Unfortunately, a conviction for public intoxication is not under the TABC, but rather the Texas Penal Code.