Summary
The Republic of Texas was formed as an independent government not joining either the Allied States of America or the Remnants of the United States of America (but later siding with them). The most likely reason why it became independent is because of the Republic of Texas militia groups that were already located in Texas and so saw it as their chance to convince the state governor to declare independence, or these movements took power themselves and brought order and help to the people and so forced the state government into surrendering to the needs and requests of their people, which formally created the Independent Republic of Texas.
Government
It is strange that Texas still only has a Governor and no President, the reason behind this is either there isnt enough time or money to waste holding presidential campaigns, or the state government thought itd be better to keep the current governor as the head of the nation so as to not cause political tensios from within and tear the newly formed government apart into taking sides, or because they are waiting for the right time to stabalize things in their nation before having a president.
Army
Texas's army is shown to be only theTexas Air National Guard and the Texas National Guard. It's possible that Military Units from the US have defected to the Texas Military.