The search for arrest records and warrants, whether for Calhoun County, Texas, or any other county in Texas, can be a vexing, time-consuming process or it can be painless and straightforward. The difference, as with almost everything that takes place mostly online, comes from how effectively you know where to look.
The best place to look bar none for arrest records for anywhere in Texas is through the Texas Department of Criminal Justice. This organization, run by the state, makes it its business to track and maintain all of the details that have anything to do with persons that have been arrested and convicted in the state of Texas. While this means that detailed criminal histories may not be available, and that you may also not be able to find information for anyone that isnt currently in jail, it can and often is a hugely beneficial tool.
You can find the main web page of the TDCJ here: http://www.tdcj.state.tx.us/. However, the tool that will allow you to search all of the criminals in Texas, at leas those currently appearing in prison, can be located here: https://offender.tdcj.texas.gov/OffenderSearch/index.jsp .
If you are looking for active warrants for Calhoun County, Texas, then you will need to get into contact with the sheriff of that county. This office maintains a webpage at http://www.calhouncotx.org/sheriff.html. You can reach them via phone at (361)553-4646, or at the address of 211 S. Ann Street, in Port Lavaca.