My front bucket seat backs are shakey and move considerably left to right when pushed on. I checked the bolts securing them to the floor and they are tight.
Can the trouble be in the hardware securing the seat backs to the bottom part of the seats?
These are called custom buckets on the 1978 Pontiac FB Formula, but they have no special features such as reclining backs. They only move back and forward and the backs do not stay in place if folded forward.