Actually, possession of
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If there is a warrant for your arrest, then the statute of limitations is tolled (stopped.) It stops when a charge is filed. So, the case is not going to just go away.
Second, generally most states will not arrest someone for a misdemeanor warrant from another state if the person is stopped. The reason for this is the high cost of transporting the person back to the state with the warrant. (This is called extradition.)
Texas will hold the warrant forever - it will not expire. In the future, if you are stopped, if you try to get your license renewed, if you visit at the jail, etc., then they will find the warrant, confirm that it exists, and take you into custody. This can happen in any city in Texas. If the city which issued the warrant does not pick you up timely, you will be released but the warrant will continue to exist and you will go through the same process if arrested again until the time that you take care of the case.
You need to talk to a lawyer where the warrant is pending. You will need to post a bond and make an appearance in court, and take care of the case however you and the lawyer see fit (assuming the prosecutor agrees.)
Be aware that if you apply for a job and they conduct a background check, they will be able to find the open case, too.
Although I have answered the question to try to help you, you should consult with a lawyer in your area in person on the matter. In addition, my answer does not establish an attorney-client relationship between us.