You can search for your conviction record through the Texas DPS site, here, although it looks like you have to register before you can see pricing information.
You can't count on a warrant showing up through a third party service -- it may, but there's no guarantee. It's safe to say that a warrant would have been issued at the time, although sometimes a warrant that old will become inactive. You may be able to determine its status through the court in which you were convicted, although with records that old I won't promise that they'll be able to do a quick computer look-up, as opposed to pulling the record out of the archives.

