I’ve heard this before as well. Of course, this doesn’t hurt the case against secession at all, it just means that Tennessee legislators were not principled on the issue of secession in and of itself. They probably saw secession as something a state could do but that a county could not do since counties are chartered by the State.
However, I believe there are also cases where the opposite occurred. I’ve heard that when Tennessee voted to remain in the Union in the first vote, Franklin County seceded from Tennessee and joined Alabama for a few months until TN changed it’s mind (as a result of the call for invasion). Likewise I’ve heard there was a northeast Alabama county that stopped sending legislators to Montgomery when Alabama seceded and effectivy seceded from Alabama. The CSA also instituted an income tax and military conscription if I’m not mistaken. And of course their paper currency was a disaster as well.