Giuliani stays in the race. He plays up being a native son running against a "carpetbagger".
Hillary's gaffes get much more airtime now that she's facing somewhat competent - such as when a lifelong Chicago Cubs fan claimed to "have always been a Yankees fan". Or how she's "dead broke", coming out of the White House.
Have the way the Clinton's treated Daniel Patrick Moynihan come out at a bad time. If it had, it could have done real damage to her brand with NY Democrats.
She loses, badly, and decides to head back to her home state of Illinois. She runs for Senate in 2004, but loses in the primaries to some young upstart named Obama, who goes on to be President. Not even Bill can get the DNC to buy into her brand at this point, which is considered too toxic to touch.
She writes a few books, which like OTL, collect dust in discount bins in bookstores across the country, and is pretty much a punchline to political junkies.