Overall, your description of what you want sounds like a really good fit for the area. I live in Chapel Hill, and am familiar with Durham. Basically anywhere in Chapel Hill is family-friendly, and so are many places in Durham (I'm excluding Raleigh because I don't know about it, not because it isn't family-friendly).
There are subdivisions (named neighborhoods with a Home Owner's Association) and stuff in the Triangle, but many have houses with character -- I think a lot of people in the area have the same disgust for cookie-cutter houses as you do, especially in Chapel Hill
I suggest you go to realtor.com before your trip and scope out the housing scene. If you find a few houses you like, maybe write down their addresses so you can visit them in person.
I think you are well-suited for one of the younger, family-oriented areas of Durham. There are pockets of Durham filled with Duke professors and young families and stuff, and I think this will match well with your price range and desire to be near RTP. If you're raising a family, though, be sure to check which schools in Durham you're districted to -- some are much better than others, which I'm guessing you've read about in all your internet research

In Chapel Hill things might be a little more expensive and a little further away, but I think a greater fraction of Chapel Hill has un-cookie-cutter-ish houses and family neighborhoods. One neighborhood that I am familiar with, Morgan Creek, comes to mind as one that would fit that and be within your price range.
Must-see sights: Ninth Street in Durham, Franklin Street in Chapel Hill.
Good luck!! I think you'll like the area!!