You should consult with a good TX criminal attorney. I am a California attorney not licensed in TX & not familiar with their laws. In California there are reliefs that can be attempted such as a certificate of rehabilitation - if the type of violation qualifies & certain time limits have been met; and other statutory requirements have been met. Also a pardon can be attempted, if you qualify. Check with a criminal attorney in TX to see if their laws permit any methods of assistance.
This is not intended to be legal advise or as legal representation. I am a California personal injury attorney . Be aware that every state has its own statute of limitations; and statutes & case laws that govern the handling of these matters.
Depending on the facts and circumstances of the case, you may qualify for termination of registration. Texas law does provide relief under fairly limited instances. You should contact a criminal defense lawyer to see if you meet the qualifications.
Usually, you will not be able to stop registering as a sex offender. There are extremely limited circumstances that allow you to stop registering. To find out if you qualify you will have to retain an attorney to do the research and find out if you are eligible.
Answering this question does not create an attorney-client relationship. Legal advice has not been given. Also, this question and answer is posted on a public forum and therefore any attorney-privilege is waived.
It's way you late to appeal that. Unless there's some kind of basis for filing an application for post-conviction writ of habeas corpus to contest the conviction itself, you will only have the admittedly unlikely possibility of specifically asking to terminate your registration requirement. The specifics of that are complicated, but the most basic of the requirements is that you must either have been a juvenile at the time of the offense (16 or younger), or no more than 19, with the complainant close to you in age and the sex consensual. If either of those sounds like a possibility, contact a criminal defense attorney and see whether your specific situation could be addressed by the available legal remedies.