Texas Election Code - ELEC § 202.002. Vacancy Filled at General Election

(a) If a vacancy occurs on or before the 74th day before the general election for state and county officers held in the next-to-last even-numbered year of a term of office, the remainder of the unexpired term shall be filled at the next general election for state and county officers, as provided by this chapter.

(b) If a vacancy occurs after the 74th day before a general election day, an election for the unexpired term may not be held at that general election.  The appointment to fill the vacancy continues until the next succeeding general election and until a successor has been elected and has qualified for the office.