After the judge has signed the paperwork approving an adoption, you will need to get the child a new birth certificate. This takes quite long time. I've heard that it is taking as long as 6 months. So be patient.
Eventually the clerk of the court will send in paperwork to Austin, Texas Bureau of Vital Statistics. This can take up to 6 weeks since the clerks wait until they have a large stack and the forms all get mailed to Austin at one time. You have no control over how the judge's clerks handle their correspondence with BVS.
You can get a certified copy of your decree signed by the judge. It costs approximately $1 per page for a certified copy. Again, it takes awhile for the papers to be scanned into the Harris County Clerk's computer. You usually cannot get the paperwork the day that you get the adoption finalized - but sometimes you can - you just never know. If you are in a county outside of Harris County, each county handles certified copies differently. You will need to talk to judge's clerk or an employee of the district clerk's office to find out how to get a certified copy.
Fill out the proper forms off the Bureau of Vital Statistics website. Send the certified copy of the court order along with $62. Their mailing address is P O Box 12040, Austin, TX 78711.
If you have any questions, you can call the BVS office in Austin. Hopefully they can help you. When I have called them in the past they have been very helpful.