Let's get back to basics, shall we. "Does morality exist without God?"
If you don't believe in God's objective morality then you believe in a subjective morality. Subjective morality states that moral truths apply to the subject, the person, not the object. Objective morality states that moral truths apply to the object, like murder, not the subject. If murder is an objective moral wrong then it applies to everyone, whether they believe it or not. if murder is a subjective wrong, then it only applies to the subject, or person, who believes it's wrong. Subjective truths are based on our preferences and can change according to our whims. Objective truths are realities that we discover and cannot be changed by our whims.
Moral Relativism is a type of subjectivism. It states that moral truths are preferences, like our tastes in foods, or cars, or the the type of people we are friends with. What's the difference between a moral relativist and a person who admits that he has no morality at all? There isn't any. How does a relativist make a moral decision? He decides for himself whatever he thinks is best. How does someone with no morality know haw to act? He decides for himself whatever he thinks is best. If you believe morality is determined by subjective preferences then you give up the possibility of making moral judgements about anyone's actions. Relativist can't accuse anyone of wrongdoing.
Moral objectivism states that moral rules are true regardless of whether anyone believes it. Moral objectivist believe that moral rules are self evident, the same way that math and logic is self evident. We don't invent morality, we discover it. Morality is universal, applying to all people at all times. If murder is wrong for one person then it is wrong for everyone. Christians believe that God has given us this objective morality.