Is there a deadline to appoint a replacement once an elected official has resigned?
No. There is not a deadline. Section 87.041 of the Local Government Code gives the commissioners court authority to appoint to fill the vacancy with a person who will serve until the next general election but does not impose a deadline to do so. However, note that the resigned officer remains a holdover officer and therefore is entitled to salary and benefits of office until their successor is qualified.
(As of June 2019, this answer is different for commissioner's vacancies in counties with a population of more than 300,000. In those counties, if the county judge fails to appoint a replacement before the 61st day after the date the vacancy occurred, the court shall appoint a suitable resident of the precinct in which the vacancy exists to fill the vacancy until the next general election. Local Government Code §87.042).