Not too far from Greensboro is Tanglewood RV Park in Clemmons. It's a county park with full hook ups(water , electric, sewer, cable and wifi). Price is $30 I believe. Paved pads. They are not level though. They did slope (some a lot more than others), but it's where we go when we go to that area. If you call them, they will guide you to one of the better sites. I was also told that if you make reservations on line that when you put the type of unit (MH vs trailer) that it will will not allow you pick one of the more sloped ones if your in a MH. Not sure if that is true. I pick the site since I know which ones are the better ones.
Hagan-Stone is a Guilford County Park just outside of Greensboro that we've stayed at a few times. $20/night for water & electric. Nice quiet rural park.
Museums of Science & Natural History in Raleigh. Pullen Park while there (just because I used to be the engineer for the kids train twhile a student at NCSU back in the early 70's.).
In Raleigh, stay at the state fairgrounds... full hookups, easy to get into level sites.. clean, but no extra amenities for reasonable rates.. Stayed week of Christmas last year, while visiting daughter and family in nearby Holly Springs. Would stay there again without hesitation.