I have not personally visited the North Carolina Zoo, but from reviews, commentary, and pictures in the gallery here at ZooChat it seems like this zoo has an excellent reputation for large, naturalistic enclosures for African and some North American species. Their elephant and antelope/rhino exhibits in particular seem to be very highly regarded. The Nashville Zoo likewise gets very good reviews here for the exhibit criteria that you mention. Disney's Animal Kingdom has perhaps the largest, most naturalistic exhibits for African species anywhere. These zoos seem geographically closer to you than Omaha. On the West Coast I would recommend Woodland Park Zoo, San Diego Safari Park, the Living Desert in Palms Springs, Monterey Bay Aquarium, Arizona-Sonora Desert Museum and Oakland Zoo as good zoos to see large, naturalistic enclosures. The San Diego Zoo and Los Angeles Zoos are also quite good, and getting better.
I have been one of the critics of Omaha Zoo here. They have a great outdoor giraffe exhibit and a decent gorilla exhibit, but other than that I would not ascribe "large and naturalistic" to their exhibits. The Lied Jungle and Desert Dome are superficially impressive as structures, but shameful to hellish as animal exhibits. Similarly the bear and cat exhibits contain an impressive species list, but a low-quality exhibit presentation. Omaha Zoo is a zoo in transition...they seem to recognize from their master plan that they have some major problems to fix (like building a new elephant exhibit, bears, cats, etc.), but it is going to take a long time to fix.