Although the resort itself doesn't have multiple courses, the Grove Park Inn in Asheville, NC is one of the great family resorts in the U.S. The historic hotel is something special, as is the Biltmore Mansion nearby. There is hiking, sightseeing, and the town of Asheville is delightful. Make sure you stay in one of the historic rooms in the hotel, as opposed to the newer more contemporary wings. Incidently, the Grove Park has, as I understand, one of the largest private collections of period mission and arts & crafts furnishings in the U.S.
The golf course is a classic Ross layout, restored in the past 6-7 years or so. I visited a few years ago, also mostly for the golf, when our girls were of a similar age - and we all loved it. There are several other golf options nearby as well.