When a nation-state first gains independence, in the absence of its own currency, it usually declares, unilaterally, the currency that is common to the region as its official currency. This is called an informal currency union. In the early days of independence, doing so provides for economic stability as consumers and businesses can continue to transact business in exactly the same way as they always have.
Where the U.S. dollar is concerned, its status as an international reserve currency has made it attractive to countries that have no desire to adopt their own currency. Many self-governing countries even allow the U.S. dollar to circulate freely in addition to their own currency.
According to a 2014 article on the website Quartz:
“The US dollar is the most widely used currency in the world, with many countries employing it as an accepted alternative to their own currency. But some have simply adopted the currency as their own, notes and all, in what is known as “dollarization.” They don’t have control over the currency—only the Federal Reserve in Washington sets monetary policy.”
To be clear, this can be done without the blessing of the United States, as it has been in Ecuador, East Timor, El Salvador, Marshall Islands, Micronesia, Palau, Turks and Caicos, British Virgin Islands, and Zimbabwe.
If Texas wanted to have a say in monetary policy and still use the U.S. dollar, it would have to negotiate a formal currency union with the United States. Formal currency unions are common in the world; in fact, there are more than 20 official currency unions throughout the world. While a negotiated currency union with the United States would be desirable, the terms under which such an agreement could be executed likely would not give Texans any more control over monetary policy than we have now.
The most likely scenario is that Texas will adopt the U.S. dollar as its official currency in the immediate aftermath of a Texit vote to encourage stability while seeking a negotiated currency union with the United States. Depending on the terms of any negotiated agreement or in the absence of one, Texas will want to explore moving toward a currency of its own as soon as possible. Given the lack of long-term financial stability in the United States due to the exploding national debt, a Texas currency should come sooner rather than later.