I think they were sold at Hudson's Department Store.
The Cadillac and LaSalle badges have to do with the explorers, not cars. The Lincoln badge is for Abraham Lincoln. They were sold by many bike shops, hardware stores etc. For instance, Barnard's Oak Park Cyclery in Chicago sold Packard badged Schwinn built bicycles. Both Barnard's Oak Park Cyclery and Grose's Bike Shop sold LaSalle badged Schwinn built bicycles. The bicycle dealers that were supplied by the Chicago Cycle Supply Company sold bicycles with any of their many badges. I know of at least 12 badges they had, most likely more than that. Barry