-Durham Thur 8/28-Sun 8/2
Durham has regular random draw doubles at Valley Springs on Thursday evenings. I believe they usually start at 6:30. If you are playing only one course, and it has to be in Durham itself, that's the course to play (depending on what you are looking for).
Leigh Farms has had quite a bit of work done on it over the last 2 years and is really enjoyable.
Cornwallis is one of the oldest courses in NC, going back to mid-80s, and is a short putter course, but the history is there.
If you are willing to expand to the broader Triangle area, you should look at playing at Diavolo in Cary. It's not the #7 course in the world (UDisc inflation is real) but it's definitely an excellent course.
The UNC course in Chapel Hill has random draw doubles on Saturday morning, I believe at 7 AM. That's also an excellent course. It's changed a fair amount recently, which some people like and others don't, but it's definitely worth playing.
Past that, depending on how far you think is a reasonable drive from Durham (and where in Durham you are staying) there are quite a number of courses to play. You could go out as far as Cedarock Park and play the figure eight out there, which is 36 holes. It can definitely get crowded, but it's also a historic experience.
