The answer to this is driven by the facts of the case. For example, were the kids with her, has she done this before, has she developed a drinking habit since the divorce? Modification can be granted based on material and substantial circumstances that have occurred since the divorce, such that modification is in the best interests of the child. I have been handling custody cases for 19 years. I have an office in San Marcos, and I'd be happy to discuss it with you. Feel free to call.
Todd Steele's response to your question is for general information purposes only. Nothing in this response should be taken as legal advice for any individual case or situation. This information is not intended to create, and receipt or viewing does not constitute, an attorney-client relationship.
You can certainly ask the court to amend custody. Whether the court will actually do it will depend on the facts, to a degree, but equally important is your particular judge's attitude toward DWI. I have seen judges treat DWIs as if they are the end of the world, and I have seen judges who treat DWIs as no big deal. You should consult with an attorney who is familiar with your judge. If your case is in Travis County, it may be difficult to pinpoint who the judge of your case is because Travis may be one of the few counties that operates on a rotating basis. I'm not certain about this, but it is a consideration.