Is there anyway to remove a felony conviction from my record if it cannot be expunged because it was not deferred?

Felony convictions are very hard to get off of your record.  You cannot get a felony conviction expunged.  A conviction is a prison sentence or a strait probation where your sentence was suspended.  If you were not convicted, but instead, placed on deferred adjudication, then you can petition the court for an order of non-disclosure.    If you have a conviction and you were not granted a deferred adjudication, then your options are a bit more limited-- but you still have a couple.

The first is as you mention-- ask the governor for a pardon.  In Texas, this is also known as clemency.  The Texas Board of Pardons and Parole accepts these applications and make recommendations to the Governor.  Here is a link to the site which has the form you would need to begin the process: http://www.tdcj.state.tx.us/bpp/exec_clem/exec_clem.html   .  Pardons are possible, but rare in Texas.

Another option is to file a post-conviction writ.  Often, writs are a long shot, but the appellate courts have gotten much more liberal in recent years at granting writs.  Writs are like an appeal, but instead indirectly attack a conviction and try to get it overturned.  For example, if you were not properly advised of certain consequences, then a plea bargain and subsequent conviction could be undone.  Many convictions have been overturned in the last couple of years because illegal immigrants were not properly advised of deportation consequences.  Others have been overturned on new DNA results or newly discovered evidence.  All of the recent cases that overturned convictions were long-shots when they got started, but have now become more routine.  This may not work for you, but if you really want relief, take the time to have a good criminal appellate attorney review your case for possible errors which would support a successful writ.