I would recommend the US77 option as well to avoid the interstate through Austin & San Antonio. I have started routing around the big Texas cities if possible on, as BB_TX says, "2 lane highways with lower speed limits and small towns." It may take a little longer, in most cases, but it is much less stressful. All too frequently, there is construction or traffic incidents on the big city freeways which cause you to lose the advantage you supposedly gained by higher speed limits and freeways. Add to that my frequent observation of downright reckless driving by folks zipping around between lanes in autos, tailgating, or using cell phones. I don't route around big cities as much outside of Texas since I am less familiar with the lay of the land and secondary roads are not generally as good.
The road to Rockport that doxiemom11 mentions is US181 at Sinton, then TX35 at Gregory, I believe.