Trying to test my memory, I believe the Chrysler products all had them until 1964 or 1965. Seems like I recall a '64 Valiant still had the push button. They were gone, just like the fins, by 1967. I used to drive a '67 Belvedere with a 318 4 barrel, no push buttons.
Lucky you..I drove a 1961 Studebaker Lark from 1964-68..my first car..still have fond memories...