My theory is that to be eligible to be invaded by a darkmoon you have to be indicted, either as a DW or a forest hunter, and once that happens there is no turning it off. Now, I haven't tested this 100%, but I did kill an NPC on my last go around and hung around Anor Londo for an hour and never had a single darkmoon invasion, only darkwraith. This led me to believe two things.
1) Sin is not tied to your gamertag as I once thought
2) Sin (for DM invasions) is only applied if you are indicted.
I'd love to hear some other people's thoughts on this seeing as it's one of the more popular covenants.