With the caveat that I do not practice AZ tax law, let me try to provide a general answer. In an S corp, the corporation itself is not taxed. All profits and losses are passed through to the shareholders. Yes, you are insulated from liability for things like tort claims and contract claims, so long as the corporation is adequately capitalized and run properly. However, in this case, the corporation owes the tax--or at least the state says it does. That financial liability passes through as a deduction for the shareholders. Generally speaking, taxing authorities can do things that others cannot, like reaching shareholders for a corporate tax obligation, at least in a small, closely held S corp.
Certainly you need AZ tax counsel familiar with your Transaction Privilege tax and the legitimacy of the state seeking to recover from you personally.