No. Often we are tempted to leave it at that when we have to make a negative response to a question, but we always try to at least make some explanation. Sometimes our only answer is we don't know. In this case, we have to say which comes first, the grass or the tree? Covering tree roots with more than about an inch of soil will be suffocating to the roots and they will promptly push up again. Most tree roots are near the surface because of the need for moisture and also the gas exchange of oxygen and carbon dioxide at the surface. Covering them with grass sod, same problem. Stripping out the roots, worst. The roots are down there to gather nutrients and moisture from the soil. These substances are passed up stems and trunks to the leaves which, utilizing sunlight through the process of photosynthesis, manufacture food for the whole plant, transporting it back down the trunk or stem to the roots, where nutrition for the whole plant is stored. Those roots are where they need to be for the tree to survive.
In addition, we suspect that part of the problem for your lawn grass is, as you mentioned, the drought conditions and part is the shade from those trees. There are few lawn grasses that can tolerate much shade, and those that can are mostly water guzzlers. It would be your choice, of course, but we are all in favor of the trees.
We would suggest you consider putting something else beneath those trees and perhaps embark on a process of xeriscaping. From eartheasy, here is an excellent article on Xeriscape. Obviously, you do not have to do every single thing suggested for xeriscaping, but you can start small and work your way up. Without knowing exactly what else is going on in your garden, we would suggest covering the offending roots and bare ground with a nice layer of mulch. Please read our How-To Article Under Cover with Mulch.
A good quality shredded bark mulch will make a nice cool surface for the ground, sheltering the tree roots from heat and the sun, discouraging weeds from sprouting and preserving moisture in the soil. It will tend to scatter or decompose, sinking into the soil and making it healthier, over time, but it's an easy fix to spread some more on the area. And it doesn't have to be mowed. We had one letter from a homeowner this week that said they were so over grass, and we feel, in this hot, dry climate, that may be a very good idea.
Follow this plant link, Fraxinus velutina (Arizona ash), to our webpage on it to learn more about its needs and growing conditions. You might also be interested in the fact that, according to this USDA Plant Profile Map of its distribution, that the Arizona Ash is not even native to the southeast area of Texas where Harris County is located.