Is it evergreen? Prunus lauroceras is an option.

No not P laurocerasus. That doesn't have that veination or texture. Bottom left hand branch has tiny greenish flowers.

Are there any Buckthorns in your area? Rhamnus (or Frangula, maybe.) That's what I'd say up here but I don't know which ones you'd have in Texas.

Hmm, an Alder Buckthorn, perhaps?

^ Hmm, Carolina Buckthorn - another good possibility! I'll try to get a closer look later... Thanks!

Ok, here is the same type of shrub, I believe - but currently bearing red berries about ~3/8" in diameter...if this helps verification?