There is no real point of dispersion. No one thing changed. What is different is that when the Civil War broke out, Texas seceded but as its own country. Texas does not join the Confederacy. Instead, Texas claims that the United States illegally annexed Texas, and thus Texas has the right to declare independence.
Texas States

The twelve states of the Republic of Texas, as well as the counties. The Capital District of Austin is in Black, located between the States of Bryan and Crockett.

Early History of Texas

Sam Houston, as Governor of Texas, leads his people to Declare Independence in 1861, at the outbreak of the Civil War. He does this by claiming that the Annexation of Texas was illegal, and takes advantage of the fact that the Union and Confederacy are both too occupied to focus on keeping Texas. In 1862, a Constitution is made, which includes the division of Texas into 12 states, each named after a key person in the history and Establishment of Texas. 

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