The Constitution ends with "In the Year of our Lord" - who is Lord? The Constitution embodies the principles from the Declaration of Independence - these two docs go hand in hand. Check the bill of rights - we know that the founders believed freedom to be an unalienable right from God.
You can't impose "your own set of moral absolutes" because there is only one set and it is from God, not man. Your point about human evil merely strengthens my argument about the purpose of the moral law - to drive people to Christ. All men have blown it - this is absolutely consistent with Christian teachings. The institutions of mankind are flawed at best and brutally evil at worst. This does not disprove the existence of moral absolutes or a universal moral standard, it merely serves to show that no man can perfectly live up to them. You are looking at governments, try looking at some individuals and see their integrity and honesty and compassion - Washington for one. There is a clear difference in the behavior of an avowed relativist (Marquis de Sade) as compared to a moral absolutist (Washington). How do you explain that? There are men who are moral and do the right thing most of the time - Washington was one of those, and so were many of our founders.