I have a 1930 Model A with the oval speedometer. W
Both sides of the amp meter are always hot, makes no difference if the key is off or on.
If the light is staying on all the time, the problem would be in the dash lamp. Check the function of the light, when you push in and twist the lamp cover it makes contact with the bulb. Try turning the lamp cover counter-clockwise and the bulb should release from making contact.