I changed my stock shifter for a Dual-Gate in my 69 years ago, so I am kinda fuzzy on how the stock one mounts. I know the trim piece(chrome with woodgrain sticker on 69) that covers the shifter is held in with spring clips and pops out. I think the shifter is bolted to the console under there. It is also bolted to the trans tunnel with two brackets, sounds like yours may be loose.
PAWN SHOP FIND
RESTORATION IN 1980...
"You can have it all, my emipre of dirt. I will let you down, I will make you hurt"