Umm first off Jefferson wasn't there for the writing of the Constitution so any quote you pull up from him to justify your position is crap.
Adams is correct but not in the way you are using his quote.
Governments in the ME are based on Islam and are guided almost completely by the Quran.
The U.S. government isn't run strictly according to the Old or the New Testament. But the teachings contained within the Bible most certainly guided their beliefs on free will...freedom of association and the prevention of one monolithic state run church being established the new country they were creating.
Make sure next time you understand the context of the quote you're cherry picking.
Try harder next time.
Jefferson's quotes matter because he was, intellectually, among the leading lights of the Founding Fathers and of HUGE influence among those who wrote the Constitution and Federalist papers.
As for context, the statement means just what it says...and I gave it in full context of his views on governance. Christianity is linked to western moral sensibilities over time and tradition, but has ZERO direct role in our government...and that is the meaning of Adams statement. He understood clearly, and advocated, that the United States was NOT founded on Christianity...nor should Christianity be related to government in any way other than through its impact on individual men who served in governance.
Again, god is not mentioned in the Constitution...a deliberate and pointed act by men who were themselves religious. They understood how religion and rule were not compatible...and in fact were dangerous when brought together in such a fashion. This was not to prevent an Iranian style theocracy...as you cite for some odd reason...but rather to keep the potentially oppressive effects of religious advocacy out of our Republic.