It sounds like you're aiming to grow lawn only, right? I hate to disappoint, but from what you've written, lawn is not going to grow under your trees in that kind of soil. In fact, even if the soil were better for turf, mature trees will out-compete a lawn, and it would always be struggling. I think your best bet is to consider some beautiful and creative landscaping with other types of groundcovers for shade, some paths, and perhaps a seating area. Turn your lawn frustration into a lovely outdoor entertaining area. Do not lay concrete or lots of stone over the soil around your trees--you will suffocate the roots, and eventually lose your valuable trees. I lived in Houston for a number of years, and I know how precious shade is in summer. You may have a gem waiting to be polished in that yard--it just doesn't have a lawn on it.