Keep the trees- large island with shade plants and or ground covers is the answer. I’m in coastal Georgia and asiatic jasmine would be our go to. Just no English ivy!

Xeriscape. We are having more droughts every year throughout the country.

Get a soil test done for $50 or $100. If there's something wrong with the dirt, it's better to know now than after spending money on new plantings.

Find a native ground cover.

With those two beautiful trees, you will never get good grass. Mulch the area, plant with semi sun loving flowing bushes, and ground cover. As you are mulching and planting remember to put some paths through the area. It makes it easier to get into the garden to do work on it and it adds interest to the garden.

We have concentrated on creating a butterfly garden and have planted all kinds of flowers and bushes that attract butterflies and bird. Don;t forget to add a bird bath, bird feeder and a humming bird feed.

If they are healthy, do not remove the trees!

Ground cover or shade loving plants are your best bets.

Your local nursery will have lots of suggestions!

My first thought would be to hire an arborist , they appear to be in need of some pruning and shaping .

Yes, get the trees in shape--they're beautiful, dig out a bed, enrich the soil, and come up with a design plan for perennial shade plants with your garden center.

Here's a Bob Vila article about planting under trees. Note especially that beds must be dug out not placed on top of lawn:

https://www.bobvila.com/slideshow/the-dos-and-don-ts-for-landscaping-around-trees-52681

I live in Dallas, Tx too. I am not a professional landscaper just a home owner who has had the same problem as you. We originally had Bermuda grass. It didn’t do well under our big oak trees. So we planted Saint Augustine grass. We had our trees thinned out so the light can come thru. The grass is thriving now. Also one thing I’ve learned when planting sod, is to use plain (no sandy loam added) dirt before laying down the sod. Also, a sprinkler system is a must so if you don’t have one good idea to get one first before planting new sod. I hope this helps good luck!

I was also going to suggest St. Augustine grass!

native plants are so much better than water-hogging sod, especially in the face of climate stresses!

I agree you need an arborist. I can see what looks like a large dead limb. I think your could do a beautiful shade woodland garden. So much more interesting than grass. Ferns, hostas, vinca minor, and other plants, like hydrangea, will be such an asset.

Krumlauff-Weidemann Woodland Garden · More Info

Krumlauff-Weidemann Woodland Garden · More Info