Can I get my old DWI case off my record?

The answer depends on the final outcome of the your previous DWI.  The only way to remove an old DWI case from your record is if your previous case was dismissed or if you were found "not guilty" by a judge or jury.   If you were convicted and paid a fine, went to jail or were on probation, you are not eligible to have your record cleared.

If your case was dismissed and the statute of limitation has run (2 years on a misdemeanor case) or you were found "not guilty" by a judge or jury, the legal process to remove your DWI case from your record is called an expunction.  The expunction process is governed by Texas Code of Criminal Procedure, Article 55

The expunction process can take several months to complete.  In a nutshell, your DWI lawyer would file a Petition for Expunction asking a District Court Judge to order the entities that were involved in your DWI case to destroy any and all records relating to your DWI arrest.