Been going to Emerald Isle for 40 years and now live three miles away on the mainland. Needless to say, you pay to be on the ocean, but it's a lovely unspoiled beach. My brother in Cape Carteret owns a rental "cottage" on the island, and the whole family comes for a week.
Another possibility, maybe to split your stay and bring down the overall cost, is the National Forest Service Cedar Point Campground just across the bridge from the island off highway 58. Electric hookups, water available, dump station, takes reservations. It's very woodsy, has hiking trails over the marsh, kayak launch. We often stay there just to check our systems before making a long trip.
Your kids will enjoy MacDaddy's and the Golphin Dolphin on the mainland at intersection of 58 and 24. Upscale bowling alley, mini golf, go-Karts, driving range, etc and video center, family oriented. It's at a shopping area with the best food store in the area, Lowe's Foods, as well as Lowe's Hardware, Walgreens, McDonald's, Hardees, etc.
Send me a private message if you have any questions. Enjoy EI.