Short answer: No. In Texas, a private, in-state sale between two unlicensed Texas residents does not have to go through an FFL and does not require an ATF Form 4473. Form 4473 is only required when the transfer is made by a licensed dealer/manufacturer/importer (an FFL).
A few must-know caveats:
Interstate: If the buyer lives outside Texas, the firearm must go through an FFL in the buyer’s state, with 4473 and a NICS check.
Prohibited persons: You can’t sell if you know or have reason to believe the buyer is prohibited. Texas law also makes certain transfers unlawful (e.g., to minors, intoxicated persons, or someone intending unlawful use).
Optional FFL: You may choose to use a local FFL to run the background check and record the transfer, but Texas law does not require it for private in-state sales.
Frequent/profit-motivated selling: If someone is “engaged in the business” of selling firearms (regularly selling to predominantly earn a profit), federal law expects them to be licensed—then 4473s apply. (ATF updated guidance/rulemaking on this in 2024; parts of it have been litigated.)
Not legal advice; just the general rules. If you want, I can help you find an FFL near you who will handle a private-party transfer.