I managed to fix this, but I'm not 100% sure how. I went back into my last few posts and replaced the div tags with p tags. I'm pretty sure that's all I did, and it is the case that in my theme (and a bunch of others) the sidebar is differentiated with a div tag, but I'm surprised that that's all it took to fix and, and indeed, break it in the first place. I think maybe the Wordpress upgrade made changes to the html editor such that it's adding lots of spurious div tags. Meh.
But there you go. I hope that if anyone else has this issue, it is just as easy to solve!