It's hard to see the shape of the seed capsules from that picture. But they look oval rather than triangular. So no, it's not Shepherds Purse. There are lots of white Brassicaceae and I'm not familiar with Texan species. Some sort of Lepidium, I'd say.

You have 4 species of Lepidium in Dallas, virginicum, austrinum, densiflorum, and didymum. L. didymum is much different in appearance to the other 3. It grows more prostrate and the leaves continue to grow into the flower stalk. Lepidium austrinum's stem is covered in dense long hairs. L. densiflorum and L. virginicum have stems covered with very short white hairs. Virginicum and densiflorum look almost identical when viewing their upper halves. There is a noticable difference in their basal leaves. If the basal leaves have dissapeared, you can use the keys to ID them, but you need the plant in hand to use the keys. If you need help understanding the keys, I can help.

The basal leaves aren't reliable for identification. The seedpods,(silicles) of Lepidium densiflorum are wider above the middle. The seedpods of Lepidium virginicum are widest below, or right at the middle. The seedpods look widest at the middle, so this fits Lepidium virginicum imo.

^ Wow damn, awesome detective work! Native Lepidium virginicum (Virginia Peppergrass) sounds good to me, thanks!!!