I just found this for now about the FDR conspiracy. It's wikipedia so take it with a grain of salt.

While accounts focus on Cermak and the other victims being random casualties of an attempt to assassinate Roosevelt, a conspiracy theory emerged sometime before 1999,[16] starting in Chicago,[17] asserting that Zangara was a hired killer working for Frank Nitti, who was the head of the Chicago Outfit crime syndicate. John William Tuohy, author of numerous books on organized crime in Chicago, after reviewing Secret Service records,[18] described in detail in a 2002 article his interpretation of how and why Cermak was the real target and the relationship of the shooting to the rampant gang violence in Chicago.[19] The theory is enhanced by numerous researchers, citing their analysis of court testimony, asserting that Cermak had directed an assassination attempt on Nitti fewer than three months earlier.[16][20]

The conspiracy theorists suggest that Zangara had been an expert marksman in the Italian Army 16 years earlier, who would presumably hit his target,[21][page needed] though sidestepping any issues about Zangara's progressive age and health issues since his time in the war, his short stature requiring him to stand on a jostled chair, his experience being with a rifle rather than with a pistol from a great distance, and his own statements regarding his target.

Raymond Moley, who interviewed Zangara, believed he was not part of any larger conspiracy, and that he had intended to kill Roosevelt.