I don't believe Paul is dating Lulu or Twiggy for that matter - they are all OLD friends from the 60's
Possibly had dates with Rosanna Arquette, but has been linked for sure w/ Nancy Shevell, a NYC woman (and a rich girl in her own right). (she is also the relative of Barb Walters) who has a big job with the Metropolitan Transportation Assoc in NYC. Father was in shipping/transportation and the wealth came from that i think . Recently separated/divorced.
He had been seen in Southampton over the summer, buying lingerie which he gave to her . Also were together when he came to NY around Christmas, stayed together at the Carlyle. If she is The One, he was keeping it low key til the divorce was over so as not to give ammo to Mucca.
(I have a friend who lives and breathes his life, and reads online newspapers about him in languages she does not speak after I gave her the links).