The splintering of denominations added to an impetus to encode religious freedoms into the Constitution. It meant that Deists and Christians of various denominations would agree that the establishment of a state religion was a not good thing. Although the Puritans were losing strength in the North even before the Great Awakening got into full swing, evangelistic itinerant congregations were only getting established in the South by the 1740's–and were opposed by the Anglicans and powerful political leaders who were aligned with them. So I'd say that the Great Awakening helped significantly to change the mindset by creating a parallel questioning of the dominance of the Anglican church on religious grounds, alongside an increasing resistance to the political authority of the Crown with which the Anglicans were aligned.