What happens to TSA at Texas airports?

Texas would run security at its own airports, to international standards, the way every country runs its own airport security. TSA is the United States' domestic screening agency; it does not operate at airports in other countries, and it would not need to operate in an independent Texas. Texas screens its own passengers, the United States screens its own, and flights between the two are covered by the same international security framework that already governs flights between every pair of countries.

Airport security is a national function everywhere, and Texas would handle its own

Every country provides security at its own airports. When you fly out of London, Toronto, or Mexico City, you are screened by that country's people under that country's authority, not by TSA. An independent Texas would do the same at Dallas-Fort Worth, Houston, Austin, and the rest, running its own airport security operation. This is not a gap independence creates; it is a standard function Texas would perform, the way it already runs state law enforcement and emergency services. Who provides the screening changes from a federal agency to a Texas one. The screening itself continues.

There is a global security standard, and everyone flies to it

Aviation security is not improvised country by country. The international community sets shared aviation-security standards through ICAO, known as Annex 17 to the Chicago Convention, adopted back in 1974, and member nations build their airport security to meet them. An independent Texas would secure its airports to those same internationally recognized standards, which is exactly what makes a flight from one country to another acceptable on both ends. The standard is global; Texas would meet it like everyone else.

This is how flights to the US are already secured from abroad, without TSA on foreign soil

Here is the detail that answers the worry directly. Flights already depart for the United States from airports all over the world where TSA runs no checkpoint. The way it works: TSA assesses the security of those "last point of departure" airports against the ICAO international standards, and every airline's permission to fly to the United States is conditioned on meeting TSA's security requirements. In other words, the United States already has a well-worn system for trusting security at airports it does not run, on the other side of a border. An independent Texas, with modern airports built to international standards, would fit that system the same way every other country's airports do. Texas would not be a special case; it would be a normal one.

Travel keeps working, and the document question is answered elsewhere

None of this strands a traveler. Security at the airport is one thing; what you show at the gate is another, and the live site already covers REAL ID, the Border Crossing Card, and the Visa Waiver Program in the dedicated passport answer. The security operation at Texas airports keeps running under Texas authority, the screening you experience stays familiar, and the document you carry is handled in that other answer. The flight, and the line you stand in to board it, both keep working.

The bottom line

TSA's job at Texas airports becomes a Texas airport-security operation, run by Texas to the same international standards every country meets. The United States already trusts security at airports abroad that TSA does not staff, through assessments and airline requirements, which is exactly how Texas would fit in. Security stays; the agency providing it becomes Texan.