Felony convictions cannot be expunged unless the defendant gets pardoned.
Felony deferred adjudications cannot be expunged, but depending on the offense the defendant may be eligible to apply for an order of non-disclosure which limits access to the record to a very narrow group of private entities and to the government.
You really need to get with a reputable criminal defense lawyer and discuss the details of your situation.
Answers on Avvo are for general information purposes only and should not be relied upon as legal advice. No attorney / client relationship is created by providing this answer. For specific advice about your situation, you should consult a competent attorney of your choosing.
What you can do about your record depends entirely on what happened with those cases. I'm afraid that the whole seen wars thing is nothing but some kind of urban legend--criminal cases NEVER just magically disappear from your record. If they're that old and they were either dismissed or never got filed in court to begin with, you can probably get them expunge now. Otherwise, you're only hope is that you successfully completed deferred adjudication probation and got the cases dismissed, in which case you'll be eligible to apply to the court for orders of nondisclosure five years after the dismissal date. If you were convicted, you're out of luck unless you filed for expunction after some future governor decided to pardon you (not gonna happen with Rick Perry), or you obtained relief and subsequent dismissal from filing an application for post-conviction writs of habeas corpus (again, really unlikely statistically).
If you're not quite sure what happened with your cases, and they were in Galveston County, I'd be happy to check them for you and let you know the possibilities if you'd like to send me a message with your name and any case info you have.
Events do not fall off automatically from a criminal history record the way they sometimes do on credit reports. When a person says they "have two felonies," some people mean two felony arrests, some people mean two felony convictions. If your cases resulted in convictions, there is no way to get the records expunged without a pardon from the governor. If they are felony arrests that resulted in dismissals, you may be eligible for expunction. If you served a period of deferred adjudication, you may be eligible to have the records sealed.
I am not your lawyer... yet. If you would like me to be, give me a call at 512-535-1081, or send me an email to email@example.com.