Reb Wolf: The cohabitation issue is the most relevant part of the law, as a practical matter. The only part of the law that was upheld is that which bars filing for State marriage licenses with more than one spouse. But the polygamous communities in America as a matter of practice do not file for multiple marriage licenses anyways. They simply live together in religious marriages without filing with the State. And the law that previously technically prevented that, is what was overturned.
(It should be noted that even before it was overturned that law was rarely enforced unless the person was also violating other laws such as abuse of a minor. The case that got it overturned was of a man with multiple wives that made his lifestyle very public as part of a TV show. The Attorney General tried to prosecute, even though he violated no other laws, and that got the law declared unconstitutional.)