How does it work in North Carolina when you wish to start the separation from your spouse? 

Separation in North Carolina is essentially voluntary unless there is domestic violence.  Therefore, a short answer is you cannot force a spouse out even if they are committing adultery.  The only recourse to remove a spouse from the house without domestic violence is to negotiate the move through your lawyer.  In some instances, an individual can file a lawsuit for Divorce from Bed and Board however, this action does not give the court authority to kick someone out as much as it can award possession of a home for the benefit of the children as an incident to child support.  (This topic can be very complex so a short summary of the action is not really providing enough information)

A person can leave the home at anytime as abandonment is not really important to the overall case (in most instances - speak with your attorney before you relocate).

Typically, one spouse is prepared to be more amendable to moving and it is important that you and your attorney fully discuss the personalities involved and what would be the best motivation to accomplish the goal of separating.   It can become complicated when children are involved.