Thanks for the advice DB! I ended up just wiping it down, and I went over it with a really small amount of turps which cleared up everything, and didn't change the colour.
As this is my first time applying wood stain, I experimented with some different colours for the back. I went over it with a fairly dark water-based stain, which soaked really far through and made all the small imperfections in the wood show. This was a mistake originally but I ended up liking the look, made it seem sort of old and worn in, I guess. I then went over the back and sides with a Feast Watson stain (I forget the specifics) around 12 times, and that made it much darker and more 'natural' looking, for lack for a better term. I may go over the spalted cap with a little bit of the same colour to even it out, and so the sides and back don't clash as much.
It certainly doesn't look like a professional finish by any means, but I've ended up liking it (the main thing!), and I don't think anyone else I know will have one quite like it