Was Diana in a rush to marry again? No. But was she content to be alone, just busy with her charity work and sons? No.
Diana was the kind of woman who wasn't easily alone. When her marriage lacked the intimacy and closeness she needed, she found it with other men. And after they separated as well. Khan and then Dodi, she was a woman who enjoyed the company of a man. Nothing wrong with that. But she was certainly not going to be the kind of woman to stay single and focus just on her kids and her charity work either.
I find it such an old fashioned view that a divorced woman would be 'satisfied simply with her sons and her hobbies or in her case charities' when its always assumed a single man will be out looking for a new love interest. Women have just as many physical and emotional needs as a man does.
I actually think Charles would have been married much sooner to Camilla. And that Camilla would likely have been Princess of Wales as well. The length of time had a lot to do with the insane mourning of Diana in the public and concern of the out cry if he married sooner. It took years of testing waters and having a PR person help Camilla. There was also the concern of his sons and their reaction after the death of their mother.
If Diana had lived she would have moved on. People would have seen her with her own life and likely a new established love. And she would not be this martyr on a pedestal that she was made out to be by much of the public for decades after her death. Charles wouldn't have to compete with a ghost.
I also think he and Camilla would have an easier time with the public. The public would see both Charles and Diana moved on and happy with someone else. They would both be at the weddings and events of their sons, with their own lives and happiness. And people would be quicker to accept that.