Kyle,
You've gotten a lot of good info here; I grew up in Durham and went to UNC, still get back there several times a year and play golf each time. I grew up about 5 blocks from Hillandale, and my parents (now in their 80's) still live in the house I grew up in. My wife lived three blocks down the street from me, and her mom still lives in THAT house! So to me, there is no more fun day in all of golf that playing Hillandale with my brother or some HS buddies, even though we're all pushing 60 now. It is the quintessential public course, and golf the way it used to be. You are liable to get paired up with almost anybody of any race, ethnicity, and occupation; a Dook professor, a factory worker, anything.
Hope Valley is terrific if you can get a hookup. Ross circa 1926, still original routing, beautifully redone about 5 years ago by Ron Silva.
Both UNC and Dook (correct spelling) are fine, but either/both can be missed with no problem. Dook is, in fact, far overpriced; my personal opinion is that it is more about keeping the great unwashed of Durham off the course than either revenue or course quality, but that's just my opinion. Of the two, I'd play Finley, though I have trouble recommending that.
I've played Greensboro Natl.; it's a pretty good value. Grandover is o.k., but a bit pricey for what it is. Bryan Park is pretty good stuff for a muni, IMO. I like Mill Creek and the Challenge at Hideaway Farm, but they are not really must-see courses either.
I really like Tot Hill, and you're only 55 miles from Tobacco Rd., which is basically the same distance you'd drive to Durham. I'd consider both; I think they trump anything else from the public side, and a lot of the privates.