Take 59S out of Humble. As you pass through Victoria, 59 will change to 77. You would have to exit to stay on 59. So just keep on the road and you will now be on 77.
When you get to Refugio, you will pass most of the way through the town. You will turn left on E. Empresario Street (774). Turn right on 2678. Somewhere before Baytown it changes into 136. It dead-ends into Hwy 35.
Turn right on Hwy 35 and follow that into Corpus Christi. You will come into town on the big bridge that goes over the ship channel. For destinations along the water front in Corpus Christi, stay to the left as you go over the bridge.