There is a rod that goes from the carburetor down to the transmission. When the gas pedal is depressed it pushes on that rod. The rod is hooked to a lever that turns inside the transmission that causes it to down shift. It is either disconnected or broken. Chances of it being broken is slim. Check to see that the nut is missing or tight and the lever is connected to it.

I have a TCI c4 and transferred the original shift linkage into TCI transmission before installing.
My combo was ‘65 289/two barrel/c4 from factory. All original linkage in photo.

Now late 65 and 66 used a cable that runs to gas pedal linkage.

On a 68 the gas pedal bracket fastens to the firewall. It has two arms, the top one goes to the carb and the bottom one has a cable attached to it and goes to the arm on the trans.
 The correct length cable is available aftermarket and gas an adjuster on the trans end.

I guess I wrote my response so you would have to read between the lines, sorry bout that. This is the only picture I could find of the down shift mechanism. My guess it is broken internally. The only way you can remove it is to take out the valve body and some other stuff as it removes to the inside.

Is the lever actually on the notched part of the shaft?
The down shift shaft has two flats on the threaded portion on exterior.
The lever will have corresponding shaped hole...oval.

The lever and shaft if assembled correctly, should not spin like a top.
Pictures are of a 65 C4.

In layman’s terms...that’s all I know...
Small screwdriver is pointing at down shift plunger.
Big screwdriver is pointing at gear selection plunger.

Small screwdriver is also pointing at nut inside case to remove assembly from transmission.
I had to do this so I could maintain factory linkage.

Hope this helps.
Now....wanna give me a hand and put it back in car?
Just kidding!😁