Tire finding trouble please help

I am having a difficult time finding tires to fit on my original rims of my car. The car is 1938 Oldsmobile F3 8 four-door sedan. The wheels are 16 inch but they are only 4.5 inches wide. I have called Coker, and the tire size they gave me will absolutely not fit my rims. I thought about going to a 15 inch rim that is 6 inches wide but when I look under the car especially in the rear, I can’t imagine they would fit. I have heard so many conflicting opinions between bias ply, and radial, that I am seriously overwhelmed. And the tires I do find or basically for trailer. I drive this car maybe once a week around town on its 6.5 x 16 dry rotted tires that have tubes in them. But I would like to go to some car shows that are at least an hour away. Does anybody have any idea at all what tires I can use on this car that won’t put me in the poor house?