Moving out of NC

if both parties in a divorce move to different states, do they file in the state they moved to, after proving residency in the new state? then the waiting period for that state would apply to that states rules? and not NC rules for the waiting period of 1 year and 1 day.?( waiting period =separation period.)

That is correct. The laws of the new state would apply, not the laws of NC. The year and one day rule may or may not be the rule in the new state.