You need a lawyer in Texas.
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Statute of limitations vary from state to state. You need to contact an attorney in Texas to find the answer to your question.
Each state has its own statute of limitations that govern things like this, so you'll need to know what texas law says. As an aside, paying somebody the money you owe them for a bounced check might make him happy, but it won't automatically get rid of the warrant for your arrest, or the underlying criminal charge of worthless check, or whatever they call it in texas. Contrary to your thinking, the very first thing I would do is hire a texas lawyer to take care of the warrant because if you get stopped in Florida or elsewhere you're very likely going straight to jail, even though the warrant is old and gray.