It sounds like one of the buttons has come out of the steel forks in the shifter, you will need to make sure what the problem is exactly first before starting to fix it
I had a 1962 Fury many years ago, I've owned a bunch of Mopars with both the early and late pushbotton shifters but I can not remember which way (horizontal or vertical) the buttons are in the dash on that 1962 Fury now
Sorry
You may need to remove the entire pushbutton shifter assembly from the dash to see what is wrong and to be able to repair it properly
I had a 1957 Chrysler that happened on , it was fun fixing it
The plastic edges on the button had worn out so it would slip out of the steel guides on the shifter
Let us know what you find out and do to fix it