Generally, when we talk about dispositions, we're really talking about how a case is completed or ended. In a traffic case, when we ask, "What was the disposition," we're asking about the final result of the case. An attorney in Texas can advise you about more specific info.
As a general matter, a letter of disposition or court docket should show how a case ended. You should obtain your criminal history to see what's on your record. Each case, however, has some sort of disposition, and if a case file or docket states something different than what actually happened, this might or might not be able to be corrected, but because 1989 is so long ago, you might have limited options. For further assistance, you should contact a criminal defense attorney in the area in which the offenses occurred.