Deerwood is terrific. The original was a George Cobb design circa 1960 - with a very nice setting in a mature pine/oak forest. It is a housing development course, but the corridors are fairly broad and the homes are for the most part not overbearing to the golf course.
The renovation has been done for a year, and it is quite good. Some of the sharp doglegs the course was known for have been softened, drainage and irrigation updated and modernized, and strategy infused throughout. It was a total overhaul and is probably one of the better courses in the Jacksonville area. The clubhouse is modern, yes - but I wouldn't call it great. It is new, almost with a "sqeeky clean" institutional feeling.
Depending on the part of town, other good private options might be Timuquana (near the Ortega/Avondale/Riverside part of town), San Jose (just south of downtown in the San Marco/San Jose area), Selva Marina (an older club in Atlantic Beach), Sawgrass, Marsh Landing, or Ponte Vedra Club (all in Ponte Vedra Beach). Of those, I would consider Timuquana, Ponte Vedra's Ocean Course, and Deerwood to be the best.
Good luck.