Hi,
The "override tax zone" setting of shipping methods allow your customers to use the zone selected in that setting instead of the zone of their shipping address when that shipping method is selected.
But that's not what you want I guess.
You want your customers when their shipping address set to that specific zone to not pay taxes.
So your problem is not with the shipping methods but with the tax rules.
You should configure your tax rules so that they only apply to the zones where you want them to apply.
So for example, if you want a tax rule to apply to all the states of a country except one, then instead of selecting that one country in your tax rule, you should select the different states in the "zone" setting of your tax rule.
You can do that by editing your tax rules via the menu System>Taxes.