Volume control won't go to zero

Just an addendum to what Daniel said: change all the paper caps and all the electrolytics as a matter of course. All of them. Every single one will be leaky and every single one will be causing some problem that it was originally put in there to solve because its physically incapable of doing its original job any more (be that blocking DC or forming part of a filter network or something else). Those early caps had a finite lifespan, and that lifespan ended many many years ago.

If you're worried about how it looks, restuff the original capacitors with new 'guts' - but replace them all. And importantly, never just tack a new one across the old one - cut the old one out of the circuit. Particularly important for the electrolytics. If they're leaky, adding a new one might increase the capacitance and possibly even look like its solved the problem, but it does nothing for the leak.

You may already know this, but I thought it was probably a point worth making for anyone else who stumbles across the thread.

Cheers, Steve