Okay, some more tinkering has found that this information is stored in the files places.sqlite. On the old machine, that file is around 50MB. When I look at it on the new machine, it's renamed itself to places.sqlite.corrupt (although it isn't corrupt at all). I've deleted the new file and copied the old one over renaming it, but it just keeps renaming it as corrupt and won't open it.
One suggestion I've seen is to try downgrading the Firefox version and gradually upgrading from there. Doing so would be a pain in the neck but it might be my next step, I think.