I outed my cousin as a home wrecker
How did you out your cousin?
I posted it on Facebook and told her husband about it.
Family drama is meant to be kept in the family and not in public where it can ruin reputations and alienate people. If you have a problem with a family member, talk to them in person, or better yet mind your own business.
It's all her fault. Coworkers aren't supposed to sleep with my cousins husband.
Even if you got hurt and it's entirely their fault, talking about it publicly can lead you to being sued by the people you're  defaming and it can be really costly to defend yourself. Try to delete all you have posted and try to apologize to them.