Would my gun rights change?

Your right to keep and bear arms would be protected, the same as it is now, under the Texas Constitution. Independence does not weaken it, and it removes a layer of federal rules that Texans have never set.

Your rights are already written into the Texas Constitution

Texans do not depend on Washington for the right to keep and bear arms. The Texas Bill of Rights protects it directly, and those protections are already in force, applying to every Texan, untouched by independence.

The federal layer is the part that changes

The gun rules made furthest from Texas are federal rules. An independent Texas would set its own firearms policy, answerable to Texas voters, rather than having it set in Washington for fifty states at once.

The bottom line

Your gun rights are protected by a Texas Constitution that already guards them, and independence puts firearms policy in Texan hands. See the dedicated carry answer for specifics.