You can either change the behavior to align with the attitude or reject your belief to eliminate the conflict between attitude and behavior. There are no evidence to suggest that one method is more likely than the other.
hmmm, I remember learning that people tend to change attitudes rather than behavior, but I don't have a source for that. It is worth noting that cognitive dissonance does not have to be a difference between an attitude and behavior, but could also be between two conflicting beliefs or values.
The theory is such that you change your attitudes to match your behavior. If you find yourself acting or having acted in a way that goes against a belief you held, it is easier to modify your attitude to be in line with your behavior rather than be in conflict. The cliche phrase for this is "justify your actions".