I'm in a similar area as yours. Have been looking into doing "hugelkultur" beds and "keyholes" underneath crape myrtles and planting fava beans (nitrogen fixing) around mulberry trees among other plans that focuses on perennial vegetables. Besides this, mulch. LOTS of mulch.

The oak tree that my veggie patch is near was there for decades before I decided to remove some lawn for this bed. I chose to put the bed where it is, to the north of the tree, so there would be shade during the hottest part of the day. After working with it for a couple years, I'm really liking it, and from what I've read from others, the results are very good. Tomatoes are still producing after being planted in March, and last year I was still picking lettuce after people started talking about bolting. Some of the things I tried there need more sun, like Okra, probably sweet potatoes, some of the beans, but as a recently transplanted yankee, I haven't given myself too much grief over these incidents, and will do it differently yet again next year, always learning.

The tree does rob the area of moisture but, like you said, so would more mid-day sun. Also have the dry spells you mentioned, weeks without a drop of rain. I spot-water with the hose and have very few weed problems from allowing the space between plants to remain dessicated. Although that is less of a factor as I keep adding OM, the soil throughout this area is becoming much more moisture-retentive, dark, humusy, easy to dig in. It's under about 18" of leaves for now, everywhere there isn't still something growing.

I'm a huge fan of mulch but find it gets in the way when trying to germinate seeds, and have stopped putting mulch in areas where I want to grow from seed. Using just compost, leaves, and grass from the mower bag seems to work better in these areas.

No idea what tree to suggest for TX, but would hope you'd choose some kind of native if not a specific edible. You may also want to investigate espalier techniques for your tree, and various arbors/vine supports. As well as having a purpose of being visually pleasing and/or a supports for vines, they make little spots or rows of shade.

It's a good time to lay some cardboard to smother the grass/weeds for your new planting area, cover with whatever organic matter you have, leaves, kitchen scraps, mowed grass, start improving the soil right away.