Keep on searching for a bank that'll take you, keep on saving and you'll get that down payment. Houses are cheap down there (at least compared to Houston) and as long as you purchase with the house as an investment property in mind you should be able to do alright. Insurance is a killer on the gulf coast, if you're looking for purely an investment property you may need to look further inland. Make sure to check your flood maps.
I'm guessing you work at the port or the refineries. You'll get **** for the Prius or whatever discount car you end up with but it'll be worth it. I'm in the oil field and drive a crappy but reliable little no frills car but have steadily been building my side hustle. That payment isn't worth it.
You don't want to put your future family in a bind then put your focus on what you need to do now. I was working offshore half the year, walking to the office when I was onshore and living in a $400/month apartment stacking money because I knew I'd be supporting a family some day. Got the family now and we're in a much much easier position because of it.