Graham per se discontinued passenger car production during 1941: the Skylark / Hollywood venture was judged to not have been successful. Thereafter, Graham went after defense contracts, and did reasonably well during the war. Thereafter, Joseph Frazer planned to revive passenger car production, building an eponymous vehicle. He met Henry Kaiser while looking for financial help and the two agreed to work together, forming Kaiser-Frazer in 1945. Kaiser-Frazer and Graham-Paige worked out a joint operating agreement and leased former bomber factory Willow Run, producing two Kaisers for every Frazer (the latter of which was technically a Graham-Paige product). Short version, you're mostly correct.
Probably you may be right in that they'd be another postwar independent, quite possibly the smallest save for niche market Checker. But it might butterfly away Kaiser-Frazer altogether, or yield another Kaiser venture (perhaps Kaiser acquires Willys perhaps eight years earlier?). One thing Graham would have to do is to carry out a complete restyling since the Spirit of Motion and the Hollywood would have been badly dated by 1945. Perhaps ITTL Joe Frazer hires Howard Darrin as he did IOTL and Darrin re-styles the postwar Graham, including a coupe and a convertible. Perhaps that total restyling would give Graham enough of a postwar boost to gain sufficient momentum to carry the company into the mid-1950s: one might envision a Studebaker-Graham-Packard merger with Studebaker challenging Chevy / Ford / Plymouth, Graham about on a par with Oldsmobile / Mercury / Dodge-De Soto, and Packard with Cadillac / Lincoln / Imperial. That might have survived until the 1960s and perhaps longer if that organization had acquired Willys along the way.
One thing I just found out: the Skylark / Hollywood project was also to form the basis of a revived Franklin Olympic. Imagine an air-cooled mill under that iconic coffin nose...