It's the tee handle style with the big black button, right?
The button presses down on a cable, or linkage, that is inside the chrome shaft and when you press down on it it allows the tab at the bottom of the linkage to clear the detent and allow to change gears. In the attached picture you can see there's a spring below the "tab". Yours must be "frozen" or "siezed" up.
I am not sure about the 67 Camaro shifter being the same, but it may be.
It's real easy to remove the shifter handle ... just take off that nut you see at the base of the shaft where it attaches to the gray thingy in my picture (I don't know what to call it. So take apart and shoot some WD40 in there to free it up. Depends onwhats wrong with it ... it may or may not work.