What moral guide should American society use for making moral decisions, moral absolutism, moral relativism, or something in between?

It's a mix: Some basic founding values supersede everything, but our government thrives on finding compromise between people who disagree. 

The U.S. Constitution lays out some ground rules and takes some clear moral stances. However, it also protects peoples' right to choose and practice their own system of belief, which limits the extent to which we can impose a moral code as law. 

Our system of government, which is called a "federal republic" is shaped around this protection for diversity of moral codes by mostly favoring the majority opinion while protecting the rights of those in the minority.

The Constitution says that people should have the freedom to practice and express their own beliefs and opinions. This includes the freedom to practice any religion. It also includes the freedom to speak, publish or otherwise express thoughts and opinions. Groups who share beliefs are allowed to gather peacefully. 

In other words, while we can make laws that are consistent with some religions or moral codes, we cannot force others to practice one religion instead of another. For example, at least two of the biblical Ten Commandments also appear in U.S. law in some way: "Thou shalt not kill" and "thou shalt not steal." However, making a law for the second commandment, "Though shalt have no other gods," would violate people's constitutional right to freedom of religion. 

The Constitution does hold certain moral ideals, most of which pertain to what are now considered human rights: That people have a right to life, liberty, security of person (not being wrongfully imprisoned or tortured), freedom from slavery, equality before the law and other guarantees to protect the basic humanity of each person. 

Our society has to evaluate its values on a daily basis and of course, not everyone agrees. Individuals' interpretation of the Constitution, religious beliefs, moral codes, opinions and life experiences may lead them to draw very different conclusions about what is right. The morality reflected in our laws continues to evolve not only as societal values shift one way or another, but simply because we constantly face new questions. 

At times, Americans have disagreed with the moral behavior set out by the Constitution so strongly that they changed it. That is done by making "Constitutional Amendments," which are extremely difficult to pass: The legislatures of 3/4 of the states (38 states) need to approve the change.

In fact, the protection for freedom of religion and peaceful assembly was the first of these amendments. Others that pertain to morality have included abolishing slavery, universal suffrage (the right to vote regardless of race and later, gender), and abolishing—then reinstating—the right to make and sell alcohol. 

The Declaration of Independence expresses that the founding principles of the law—particularly individuals' right to "life, liberty and the pursuit of happiness"—should be prioritized over everything else. This assertion that each individual matters, deserves equal treatment and has the right to live freely is a moral position that we have used to guide our decisions as a society so far.