Old three tab roof with spongy decking Powell, TN.
Spongy decking can be caused by a lot of factors. Usually, the reason is that the rafters may be too wide, around 24-inch wide rafter spacing and the deck boards might be too thin like less thin than 1/2 inch.
The fact that poor roof ventilation can cause boards to delaminate or the glue in plywood or OSB to separate and go bad will be a reason for these boards to be spongy.
This would mean a complete reroof of this project, new boards, and new shingles.
Aged plywood has deteriorated and some sheets will need to be replaced.
No soffit vents. Consider powered vents.