"... people would have to pay higher taxes to their state and local govenrments too ..."
Not so since state and local governments now pay federal taxes in the form of what they pay their employees and their suppliers. In other words, the governments tax the,selves presently and would continue to do so. Trying to remove this amount of taxation makes no sense since the income tax presently taxes government - and the various flat tax proposals also. Not doing so under the FairTax would amount to giving government a tax-free advantage to compete against private firms - something that no one should wish to see.
Your post is simply untrue, under the FairTax the costs of state and local government is going to go up substantially.
Things that they buy will have the FairTax (currently stated at 29.87% exclusive) added to them and remitted to the Federal government. Since pre-tax prices are not going down anywhere near that much (more like 8% MAX), this will mean the cost of aggregate purchases will go up by at least 19-20%, and probably more since the tax rate is artificially low.
And since you claim government employees keep their same take-home salaries, these will have the FairTax added and the only savings is the removal of employer half FICA, so cost of salaries (non teacher) will also go up at least 20%.
This extra 20% will have to be raised through new additional state and local taxation as a result of the general FairTax-induced price inflation.