Here is a list I found of Italian restaurants in Wilmington, which might help you a bit, in at least knowing what is available:
Wilmington Italian Restaurants
Personally, the non-chain restaurants I have eaten at are:
Michael Anthony's and Elizabeth's Pizza. Also, it's been awhile, but unless it has changed hands and names, I've eaten at the Terrazzo Trattoria on Military Cutoff.
The Trattoria's food was good, as I remember. But it seemed to me to be more of a cafe, with tables outside. I don't think it had the quiet ambiance you seem to be looking for, but I may be wrong.
Elizabeth's Pizza and Italian Restaurant has good food - eaten there many times. However, it is almost always very crowded and busy. You don't go there for the atmosphere.
I ate at Michael Anthony's several years ago, before I moved out of the area. The food was very good, as I remember. It also seemed to have more of the quiet atmosphere. Can't speak for the place currently though, but you might give it a try.
If it were me, I would not think it too far to drive to go to the Little River/N.Myrtle Beach area, where is located the best Italian restaurant I've ever eaten at: Benny Rappa's Trattoria - very small, wonderful candlelit atmosphere, all the food made fresh by the owner, menu written on a chalkboard on the wall - changes daily. I've eaten there many times, and it was always fabulous.