I am about 5 months older than the (not so) Great Pretender, and I registered right after my birthday in 1980 (if my wrinkled-up old brain remembers correctly. I will look for my card and report back, or else order it from SSS if not.
We must be about the same age(4/61). I remember going to the post office after my birthday and filling out some forms, and I guess I got some kind of card afterwards. The draft registration was big in the news at the time, but by this time the all volunteer military was starting to take hold.
I really doubt my name and address are sitting in folder in a government office. It is probably long gone.