As the wife of HRH the Prince of Wales her proper title was *HRH the Princess of Wales * (without any mention of her Christian name). After the divorce she was allowed to use the title Diana, Princess of Wales, without the HRH.
Any title along the lines of *Princess Diana * would suggest, incorrectly, that she was born into the royal family e.g. Princess Anne (now known as the Princess Royal), daughter of Queen Elizabeth II, or Princess Margaret, daughter of King George VI.
For the purposes of American newspapers, before her divorce, “Diana, the Princess of Wales,” was the standard. In that case, the commas were setting off her title as an apposite or or specifier to her name, so that you have both her name and her title given. Alternatively, you could refer to her simply as “the Princess of Wales” on first reference and “Diana” or “the princess” thereafter.