What happens to the interstate highways?

They stay exactly where they are, and Texas keeps running them, because Texas already does. The roads are not going anywhere, the asphalt does not belong to Washington in any practical sense, and the agency that builds and maintains them is a Texas agency. Independence changes the name on the funding stream, not the highway under your tires.

Texas already owns and maintains these roads

Here is the fact that settles most of this question before it starts. The Interstates inside Texas, along with the US and state highways, are what the state calls "on-system" roads, and they are built, maintained, and operated by the Texas Department of Transportation. TxDOT crews patch them, TxDOT engineers design them, TxDOT contracts rebuild them. The federal government sets standards and sends a share of the money, but it does not run a single mile of road in Texas. The day-to-day machinery of the Texas highway network is already a Texas operation, staffed by Texans, headquartered in Austin. Independence does not require Texas to take over the roads. Texas is already doing the work.

The "Interstate" label is about a federal standard, not federal ownership

The word "Interstate" leads people to picture Washington owning the highways. It does not. The Interstate system is a set of design and connectivity standards plus a federal funding share, layered on top of roads the states own and maintain. When Texas is independent, I-35 and I-10 do not get pulled up. They become part of the Texas national highway network, built to the same standards, connecting to the same crossings at the border. A road that runs from Laredo to the Red River is a Texas road. What changes is the paperwork above it, not the route.

The roads keep connecting to the United States, because that is how borders work

Highways cross international borders all over the world. You can drive from the United States into Canada or Mexico today on a continuous highway, through a port of entry, without the road changing into something foreign at the line. An independent Texas keeps every highway connection it has to the United States, the same way contiguous countries keep theirs. The live answer on the post-independence border walks through how smoothly a Texas-US border runs using the long, friendly US-Canada line as the model. The trucks and cars keep rolling across; the road on each side is maintained by the country it sits in. That is already the normal arrangement between neighbors.

Tolls, expansion, and road policy become Texas decisions

The real change is who decides. Today, Texas highway plans answer in part to federal standards, federal strings, and a federal funding cycle sized for fifty states. As a sovereign nation, Texas sets its own highway priorities, its own design rules, and its own policy on tolls and expansion, funded and directed entirely from Texas. Exactly what that policy looks like is a decision for the future Texas government, not something the movement pre-writes. What is certain now is that the authority over Texas roads would sit with Texans, which is the whole point of self-government.

The bottom line

The Interstates are Texas roads already, built and maintained by a Texas agency, and they keep running and keep connecting to the United States after independence. The "Interstate" name is a federal standard, not federal ownership. What changes is that Texas, not Washington, decides the future of its own highways.