Just my $.02 worth. That may help a bit, but have you checked to see the condition of the 2 retaining springs and "pins" that are supposed to hold all those bits and pieces together???
Attached pic captured from a Chevy Master Parts and Accessory Catalog I have shows all the parts meant to hold the shifter arm in place (see below).
I would double check on the condition/tension of the 2 springs (marked by red arrows) and the 2 "pins" the springs force into the holes on the shifter to keep it in place before trying to "squeeze" the shifter sides together.
I didn't look up individual part numbers, but could do that and add that info later if you need it to try and track down needed parts.
Dan
~ Dan 1951 Chevy 3 window 3100 Follow this story in the DITY Gallery "My Grandpa Carl's Truck and How it Became Mine" 1966 Chevelle (Wife's Hot Rod) | 2013 Chevy Silverado (Current daily driver) US Army MSG Retired (1977-1998) | Com Fac Maint Lead Tech Retired (1998-2021)
