Would the U.S. Constitution still apply in Texas?

No. Once Texas is an independent nation, the U.S. Constitution no longer governs Texas, and the Texas Constitution becomes the supreme charter of the land. That is not a loss of protection. The Texas Constitution carries its own Bill of Rights, and in several respects it protects Texans more strongly than the federal document does.

A constitution governs its own nation

The U.S. Constitution is the founding charter of the United States. It binds the states that are part of the union. When Texas restores its independence, it leaves that union, and the U.S. Constitution stops applying to Texas the same way the constitution of any country stops at its border. This is simply what independence is. Texas would be governed by its own supreme law, made and ratified by Texans, answerable to Texans alone.

Texas already has its own constitution, and it is older than you might think

Texas is not improvising a charter. The Texas Constitution has been the governing law of this state since 1876, nearly 150 years, and Texas has operated under its own constitutions since the Republic of 1836. It opens, like the federal one, with a Bill of Rights. Article 1, Section 2 declares that all political power is inherent in the people and that they have at all times the inalienable right to alter or reform their government. Article 1, Section 1 still describes Texas as "a free and independent State." This is a mature, tested charter, not a blank page.

In several places it protects Texans more, not less

Here is the part the scare stories never mention. The Texas Bill of Rights is longer and more detailed than the federal one, and in a number of areas it goes further. Texas courts have repeatedly held that the Texas Constitution can grant broader protection than its federal counterpart, and that it can never grant less. Texas protections in areas like search and seizure and due course of law are applied independently of the federal versions, which means a Texan can be protected by the Texas Constitution even where the federal one would not reach. Independence does not lower the floor on Texans' rights. It puts those rights under the direct guardianship of Texas courts, with no outside authority able to narrow them.

Section 29 was built for exactly this

The Framers of the Texas Constitution anchored these rights against erosion. Article 1, Section 29 states that everything in the Texas Bill of Rights "is excepted out of the general powers of government, and shall forever remain inviolate, and all laws contrary thereto, or to the following provisions, shall be void." That is a Texas-made guarantee, written by Texans, enforced by Texas courts. It does not depend on Washington. It never has.

The good parts of the federal tradition are already in Texas law

The protections Texans value, free speech, free exercise of religion, the right to keep and bear arms, due process, trial by jury, security against unreasonable searches, are all present in the Texas Bill of Rights, often in stronger or more explicit form. Independence does not ask Texans to give up the rights they grew up with. Those rights live in the Texas Constitution, and that is the document that governs an independent Texas.

The bottom line

The U.S. Constitution stops applying in Texas, because an independent nation lives under its own supreme law. The Texas Constitution takes that place, with its own Bill of Rights that in several respects protects Texans more strongly, guarded by a clause that makes those protections inviolate. Texans trade a charter governed from outside for one governed entirely by Texas.