This is an interesting question. So far as I know, the plumbing codes require overflows only on bathtubs and they are mute on other fixtures. Shower stalls do not have overflow drains. Nor do kitchen sinks. Almost all bathroom lavs have them built in by the manufacturer. Toilet bowls do not have them, nor to most laundry sinks.
I some cases, the codes specify a floor drain to catch overflow water (as in some bath and laundry room installations).
Since you mention that Texas requires sink overflows, I’d like to know if that means that Texas kitchen sinks must be equipped with them?