In reference to whether or not the Constitution is prohibiting intermingling of government and religion, or only Federal government and religion is somewhat unclear. As per usual, our best bet for determining it, at least in my mind, is to look at Thomas Jefferson's opinions. Unfortunately, he seems to waffle a little on this point. First, he would claim, in a letter to Samuel Miller (Jan 23, 1808) that:

I consider the government of the United States as interdicted by the Constitution from intermeddling in religious institutions, their doctrines, discipline, or exercises. This results not only from the provision that no law shall be made respecting the establishment or free exercise of religion, but from that also which reserves to the states the powers not delegated to the United States. Certainly, no power to prescribe any religious exercise or to assume authority in religious discipline has been delegated to the General Government. It must rest with the States, as far as it can be in any human authority.

So it would seem that you would be right. But, at the same time, he was the principle author of the Virginia Statute for Religious Freedom, which states:

To compel a man to furnish contributions of money for the propagation of opinions which he disbelieves and abhors, is sinful and tyrannical; that even the forcing him to support this or that teacher of his own religious persuasion, is depriving him of the comfortable liberty of giving his contributions to the particular pastor whose morals he would make his pattern, and whose powers he feels most persuasive to righteousness

So it would seem that while he believed that the Constiution delegated such authority to the states, that the states shouldn't have that power either, especially considering the fairly vehement language he uses in the statute.

So, to sum up -- maybe you're right and maybe you're wrong. Who knows?