I don't know what you mean by the "big" and "small" birth certificates. If your birth certificate is correct, all you need to do is request a certified copy of the certificate, and that is all you should need.
If you do need to correct the certificate:
You must have an older relative complete and sign a VS170 Application to Amend Certificate of Birth (pdf 58K) or a VS172 Application to Amend Certificate of Death (pdf 45K) in front of a notary public. If you are a minor, both of your parents must sign the form before a notary public to show they are in agreement to the change or correction.
Some changes or corrections require a supporting document to prove that an error has been made. See page two of the application for a list of corrections that can be made and whether the correction requires documentation. A suggested list of supporting documents is also printed on page two of the application.
There is a fee of $15 to file the amendment (correction) and an additional fee of $22 for a certified copy of the birth certificate once it is corrected. Send the completed application, along with the required documentation (if any) and the correct fee, to our office at:
Texas Vital Statistics
Department of State Health Services
P.O. Box 12040
Austin, TX 78711-2040
A specialist trained in parentage issues will review your application. However, we cannot file the new application until all aspects of your application have been reviewed. If filing is not possible, Texas Vital Statistics will send you a letter explaining the situation.
In other words, I doubt you can go to Austin and get this done any more quickly. Sorry.