I wonder whether most Americans are fully aware of the form of government our Constitution guarantees.

of the United States Constitution guarantees "to every State in this Union a Republican Form of Government...."

Republican, of course, does not refer to the political party but to the form of government.

Is this the guarantee to which you refer?

That's absolutely correct!

Our nation is a Republic guaranteed by the Constitution of the united States of America - not a Democracy.
Democracy and Republic, are not only dissimilar but antithetical, reflecting the sharp contrast between (a) The Majority Unlimited, in a Democracy, lacking any legal safeguard of the rights of The Individual and The Minority, and (b) The Majority Limited, in a Republic under a written Constitution safeguarding the rights of The Individual and The Minority.

The chief characteristic and distinguishing feature of a Democracy is: Rule by Omnipotent Majority. In a Democracy, The Individual, and any group of Individuals composing any Minority, have no protection against the unlimited power of The Majority. It is a case of Majority-over-Man (Mob Rule).

A Republic, on the other hand, has a very different purpose and an entirely different form, or system, of government. Its purpose is to control The Majority strictly, as well as all others among the people, primarily to protect The Individual’s God-given, unalienable rights and therefore for the protection of the rights of The Minority, of all minorities, and the liberties of people in general.