She definitely did date Alan Osmond for a short time. He was interviewed in one of the many documentaries made about her life. Plus I have seen some home movie footage of the two of them at her parents house.
I met her a few times and she always had a sort of wise cracking, jokey, public face and a much darker and depressive private one. She seemed to be constantly worrying about things being perfect. Apparently, she got that from her Mother who also had issues - borderline OCD and a total control freak.
The Little Girl Blue book is pretty good, although a little one sided and I'm sure she wouldn't have been happy to see her Mother and Brother 'demonised' in quite the way they are in the book.
The Karen Carpenter Story TV movie is OK for what it is. It skips over/omits a lot of events and sanitises others, as it was heavily controlled by her family. They demanded many rewrites of the script. On the other hand, they allowed scene to be filmed at the family home and even had the lead actress Cynthia Gibb, wearing Karen's own clothes and even her makeup! So in some senses it has a great deal of authenticity. Worth a look if you haven't seen it.
I think I remember at the time when she was dating Alan Osmond there was talk that they had became engaged and then the next news was that Alan's mother said that a marriage could not go ahead because Karen wasn't a Mormon. Now I don't know if I have got all this right or not?