Here is the ball end of a '38 Twelve shifter with the pin.

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This is off a Pierce, but it shows the basic idea

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I don't know your car, but it sounds like you might be missing the pivot pin. Does your car have screw on cap to retain the shifter? If so, you may be able to unscrew it and look at the ball of the shifter and see if there is a provision (hole) for a horizontal pin that would go from left to right.

Something has to limit the rotation of the shifter.

Maybe this one would work.

https://springmenders.com/

Great news. Since you can get to the valve ends you could try rotating the valve with some pliers. Just another thought

For curiosity, you could remove the pistons, which would take a lot of penetrating oil, heat, a BFH, a drill to drill into the ring seats to get kroil or something similar into the rings.

Then you can mic the bores and see how much over it is bored and have a shop do an ultra sound to determine the remaining wall thickness, or attach a chain to it and use it as a house boat anchor.

Tim, I see what you are talking about looking at the parts book. The problem I have is the solenoid bolt holes don't line up with the round starter (See rod in the mounting hole.) It had an OD tag.

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Another scammer today.

Another scammer today.

Mine is in the photo on the left and is a very early '38.