OK, we're getting somewhere! Ran a wire between the backs of the volume and tone pots and it's solved the buzzing issue, so cheers for that, guys!
However, I'm still having issues with the volume pot – even when all the way down, it doesn't totally cut off the sound. Then when I turn it up, it doesn't get any louder until right near the top and jumps up to more or less full volume. So what have I got wrong here?