Well, to be precise you are correct. But to be precise, just as they referred to "our Creator" in the Declaration, they referred to "our Lord" in the Constitution.
Just as this infers, their writings indicate that they attempted to establish government that was secular in nature, but which at the same time acknowledged God as did the People (and their God in fact, in the name of Jesus Christ who was "our" i.e., their Lord).
The Ratification of the Conventions of nine States, shall be sufficient for the Establishment of this Constitution between the States so ratifying the Same.
Done in Convention by the Unanimous Consent of the States present the Seventeenth Day of September in the Year of our Lord one thousand seven hundred and Eighty seven and of the Independence of the United States of America the Twelfth. In Witness whereof We have hereunto subscribed our Names.
Go Washington - President and deputy from Virginia
Exactly one citation, in the date of the document, does not a theocracy make, regardless of your wishes.
And Jesus was NOT the "Lord" of some of those folks - like Paine, Franklin and Jefferson, who were Deists.
The Founders were interested in protecting religious freedom, including the freedom of those who had NO religion. Selectively quoting and thumping your Bible harder won't make it any less true.