Charges were dropped, therefore no convictions and no problem.
But since you say these were recent, you may have to produce a signed and sealed affidavit as to the final decision. This would be produced from the court, or more likely the county clerk where the dismissal took place, and submit that with your application. The final outcome of a Texas court does not always get registered on the TX DPS database especially a dismissal, and even if it does, may take a longer time than you want to wait before you apply.