I'm reasonably confident those channel drains will resolve the issue.
Houston is a bit different than many parts of the country. Typical homes are built using post-tensioned slab-on-grade construction. We have no basements, or any portion of the home sub-grade (with very few exceptions). Most homes built in the last couple decades are also built up above street level at least a foot or two, as the streets are designed to act as flood channels when the storm drains and bayous overflow. So while our water table is comparatively pretty high, adjacent to the house you'd still have to dig a bit to find water. Along the same lines, if the water table were high enough to reach the slab, ground water would be weeping out of the ground at the street.
We actually have the reverse problem. If you don't regularly water the ground around your slab, you begin to have leveling problems due to the dirt drying and shrinking. You end up with parts of your house unevenly sinking into the ground, cracks in walls, doors and windows that stick, etc. Fixing it means digging under the beam of the slab and using a hydraulic press to drive concrete cylinders into the ground until you can use them to lift the house back to level. We aren't built on bedrock around here, it's all dirt and clay.