In Texas, the only place to grow grass at all is under the trees because of the water restrictions. But even 'shade' loving grass seed is wimpy in the shade. I assume you have a regular lawn? If that is doing well your former owners must have had the knowledge to 'train' that grass to grow deep roots and be drought tolerant. When you water do not do it by hand. Soak it deeply and do not water again until you see your footprints stay down when you walk on your lawn. Otherwise, you are wasting water and ruining the training of the lawn's roots. Lawns with shallow roots are the first to go 'dormant'. Lawns can survive a few 'dormancy' trials but they are vastly unhealthy for our expensive lawns. Best to reduce the size of your lawn and baby a small, well defined and healthy crop of grass.
A professional lawn always has a clear, defined edge. Curves have a definite radius and only changes when the radius central point moves to the other side of the lawn. The only edge I will ever use is a shovel made trench between plant beds and the lawn (cutting from the lawn side, 6" deep, 6-10" width, throw the excess soil on the plant beds). The human eye sees this edge as long as the bulk of the lawn is uniformly green.
It is critical to mow no shorter than 3"!! This shades the soil and reduces evaporation. Water deeply and only water when dry...1" per week, working towards watering no more than once per week (lots of other questions and answers worth your time to go purview). Grass within the drip line of a tree competes for chemicals with the tree. A pervious mulch such as crushed gravel creates a wonderful mini patio beneath a tree. Not too deep, an inch and not at all touching the bark of the tree. Get rid of any shaded areas or tiny struggling patches of grass. Be bold with clearly made edges. Do not use concrete, or rocks or bricks or foo foo plastic edging!
Try using an 'organic' fertilizer such as Dr. Earth's Lawn Fertilizer. More expensive, takes longer to see results (far more healthy for the plants) and after decades caring for thousands of acres of lawn, I was blown away. And instead of 4 applications per season, only 2 or 3 are necessary. Any grass in the shade the worst thing one could do is fertilize heavily.
All soils are good soils. The different types of soil need different management practices. Clay is incredible for holding onto water and chemicals. The less you manipulate it the better. Clay plus gravel plus lime plus gypsum plus water and manipulated makes concrete!