I've never had any success salvaging a cone. I threw them away. If the cone is old and disintegrating, maybe the thread is old and should be pitched, anyway

Buy new ones. They're no good anymore.

You might be able to slide an empty of the same size inside the bad cone.

They are meant to be used up quickly, so the plastic degrades quickly in light and heat.

Yes, mine do this also, even though the thread is still good. If the thread disintegrated as fast as the cone, the thread would have been not good in the first place since the garments that were sewn with it are still being used.