What happens to rivers that flow between Texas and other states?

They keep flowing, and they keep being shared the same way they are shared right now: by written agreement. Independence changes the legal label on those agreements. It does not change the water or the way it is divided.

Texas already shares these rivers by compact, not by ownership

No state owns a river that crosses its borders, and Texas does not own one now. The rivers Texas shares with its neighbors are governed by interstate compacts, which are negotiated agreements that divide the water between the states that share it. Texas is already party to several: the Rio Grande Compact with Colorado and New Mexico, the Pecos River Compact with New Mexico, the Canadian River Compact with New Mexico and Oklahoma, the Red River Compact with Oklahoma, Arkansas, and Louisiana, and the Sabine River Compact with Louisiana. These are the rules of the road for shared water, and they exist today, inside the union.

A compact between states becomes a treaty between countries

When Texas is independent, an agreement between two American states simply becomes an agreement between two countries. The substance of who gets how much does not have to change. Only the legal category changes, from interstate compact to international treaty. This is the ordinary way neighboring nations all over the world share the rivers that run between them. Nothing about it is exotic, and nothing about it requires Texas to surrender a drop it is entitled to today.

Texas would represent itself directly

Here is the practical upgrade. Today, when those shared-water arrangements are disputed or enforced at the level above the states, Texas depends on federal involvement to defend its share. As an independent nation, Texas negotiates and enforces its own water agreements as a party in its own right, with its interests represented directly rather than filtered through Washington. The state that has the most at stake in Texas water would finally be the one holding the pen.

Continuity is the default, for everyone

Upstream and downstream neighbors have every reason to keep these agreements stable. The farms, cities, and industries on both sides of every state line depend on predictable water. Disrupting a working arrangement helps no one. The realistic path, the one neighbors almost always choose, is to carry the existing terms forward under a new legal heading and keep the water moving exactly as it does now.

The bottom line

Shared rivers are governed by agreements, and Texas already lives under those agreements. Independence converts them from compacts between states into treaties between countries, with Texas representing itself for the first time, and the water keeps flowing.