Most building and household debris is welcome, and Texas is comparatively generous about construction material.
The state's own guidance says that brush, construction waste, and demolition debris may be accepted at every class of Texas landfill. What is prohibited at all of them is used motor oil and oil filters, lead acid batteries, and regulated hazardous waste.
The receiving landfills add their own list. Liquids of any kind, including wet paint, are out, along with batteries, tires, and anything hazardous, biohazardous, or radioactive. Asbestos material has its own handling rules and cannot simply ride along in a container.
Two local details are worth knowing. Carrollton treats appliances as bulk waste, and any unit holding refrigerant has to be tagged by a certified refrigerant technician confirming the CFCs were removed before it is set out. And the City allows fully dried paint in regular trash, including paint hardened with something like cat litter, which is an easy way to clear out a garage shelf.
Anything you are unsure about, ask your account manager before it goes in the container.