The best place to begin any kind of internet search for arrest warrants for Carson County, Texas is definitely going to be through the Carson County Sheriffs Office. This is for a wide variety of different reasons. The most important reason, however, is simply because only the sheriff will have information that you can rely on. Although there are many websites which will offer to sell you this kind of information, they may receive their information from unreliable sources, they may be out of date, or they may even be wrong.
Therefore, you should contact the Sheriff of Carson County. You may do so either at 806-537-3511, or at PO Box 972, in Panhandle, Texas.
You can also ask the sheriff about arrest records for persons that are currently in jail in the county. What this means is that you can ask if someone you know is currently imprisoned, and it is likely that the sheriff will tell you. However, for a full list of everyone within the county jail, you will need to rely on other sources.
Thankfully, there also exist official sources from which you can derive arrest records from across the state. You can find the best tool on the internet at https://offender.tdcj.texas.gov/OffenderSearch/index.jsp . As a note, this tool is managed entirely by Texas state sources and is updated constantly, so it is not a bad idea to rely on it.