People ask me, is veneer furniture a red flag or is it okay?

Depends on what's under it, a high quality veneer over a solid hardwood core, that's fine. Some very expensive furniture makers use veneer for beautiful figured grains that would be impossible or wasteful in solid wood.
Antique furniture from the 1800s is often veneered over solid wood. Where veneer is a problem, thin veneer over particleboard. The veneer is about the size of your fingernail, about as thick as your fingernail is.
It can't be refinished. Ask the salesperson what the substrate is. If they say hardwood, the furniture should be okay.
If they say engineered wood or particleboard, run, don't walk out of the store and come to Gallery Furniture. You know your answer at Gallery Furniture. Solid wood furniture, Made in America, made to last.
That's what you need in your Texas home, plus no harmful toxins emitting out of that veneer over particleboard. Come to Gallery Furniture, buy solid wood furniture, enjoy it for a lifetime.