Was American society becoming more "democratic" in the period from 1750's to the 1780's?

Yes. Society was being heavily influenced by the theme of equality and major Enlightenment ideas, such as John Locke's "Second Treatise of Government", where Locke explains that citizens have the "right to rebel" if a government is not taking care of its people, or Montesquieu's idea that a government should have "separate branches" so that one does not get too abusive.

Thus, you see the Revolutionary War and a Declaration of Independence.