Wobbly in which direction? Up/down or side/side?
Side/side can be fixed with cleats across, right?
Up/down by attaching a solid plank to the side of the existing one and screwing it in place securely?
That's probably how I would have done it. But now it's not very likely that I'll ever set foot in an attic from 1915

(unless it's about carrying a friend's belongings up 8 flights to a locked attic door) ;D