Two different remedies
Expunction destroys the record. Agencies delete their files, and afterwards you may legally deny that the arrest occurred. It is generally available where the case was dismissed, where you were acquitted, where the grand jury no billed you, or where you were pardoned. There are waiting periods tied to the statute of limitations in some circumstances.
Nondisclosure seals the record from public view. Law enforcement and certain licensing agencies can still see it. It is generally available after successful completion of deferred adjudication, subject to a list of offenses that are excluded and, for some offenses, a waiting period.
The DWI nondisclosure path
Texas changed this in 2017 and again in 2019. A first time DWI conviction can, in some circumstances, be eligible for nondisclosure after the sentence is discharged, if there was no accident involving another person and the BAC was below 0.15, with a waiting period that is shorter if an ignition interlock was used. Deferred adjudication became available for first DWI in 2019, which opened a nondisclosure route that did not exist before.
The eligibility rules here are fiddly and they turn on details of your judgment. Bring me the paperwork and I will tell you what is available.
Why it is worth doing
Background checks are run by employers, landlords, licensing boards, volunteer organizations, and increasingly by anyone with forty dollars and an internet connection. An arrest with no conviction still shows up, and the person reading it will not do the legal analysis you would like them to do.
It does not clear itself. Nothing happens automatically. Somebody has to file.