There are plenty of smaller companies in your area that would be willing to give you a chance. Your problem is that they are not spending millions of dollars in advertising for you to notice them. You are going to have to do your homework and find them. If you are serious about this career then stay away from the mega-carriers! If you want to just drive for a few months and be paid for cheap labor then go with a mega-carrier.
Personally I would recommend going with a carrier like Central Refrigerated. They are large enough to keep you busy yet small enough to still treat you like a human being. Central also has their own school of which you can get your foot in the door for no money up front or out of pocket. They have a terminal not far from you and they do a lot of business in Texas so you maybe able to get home a little more often at least for a few minutes.
Over the road driving is exactly what it sounds like. Its over the road and you will be gone for long periods of time. The absolute truth without recruiter embellishment as far as time spent away from home is going to be about 6-8 weeks at a time with 2-3 days off. This is the average for most companies don't let them tell you otherwise. You will be sadly disappointed once you sign on with them and don't see home but every two months. Your option then will be to quit or keep driving. If home time is an issue I am telling you now that this career is not for you. You get paid by the mile and you aren't going to run 4000 miles a week doing circles around Austin.
You want to be an owner operator that's great! That's a good goal to look forward to and plan for. Give yourself about two years of company driving and then go for it. During that time learn everything absolutely possible about the game, including running your own small business. That's what it is, a small business. Your office just happens to have wheels! The only downfall once again will be home time. My wife and I ran team and owned our trucks and would typically stay out for 4 months at a time. As an owner operator you must love your job and truck and going home every once in awhile is just a bonus. The exception to this would be if you owned a dump truck or something along those lines.
Anyway man, good luck to you, here is another website that will help you find some smaller companies in your area: