Yes, no cable in '67, there is a direct linkage. But on the other hand, you have a tranny that was not available in 1967, so there is not way of knowing for sure if you still have the original '67 shifter or not. Is the handle 1 piece chrome with the entire top section being a black plastic button? '67 Firebird floor shifters are exactly the same as '67 Camaro shifters.
It can also be that the shifter detent is so badly worn that the shifter is slipping. It could be that the shifter tang that is released with the button is so worn that the lever is moving freely. It could be that the shifter bushings that it rides on are totally shot allowing the lever to be loose and sloppy and therefore not riding on the shifter detents anymore. It could be that either broken springs inside the handle or gummed up grease is not allowing the button mechanism to pop back up.
2012 Mustang Boss 302 #1918, Competition Orange. FGF replacement 2006 Mustang V6 Pony, Vista Blue. Factory ordered. 2019 BMW X3 (Titled to the wife, but I'm always driving it for her. So I'm claiming it) Old projects, gone but not forgotten: 1967 FB 400, original CA car. After 22 years of work, trashed by the guy who was supposed to paint it. I had to sell it. 1980 Turbo Trans Am 1970 Mustang fastback, 351C 4Bbl, auto 1988 Mustang GT, 5 speed 1983 F-150 4x4, built 302 1994 Chevy K2500 HD 4x4, 454 TBI