I don’t have the new version (yet), however, in the older versions, what you need to do is view the workshop that the dwarf is in. This should tell you what the dwarf is demanding (which, depending on the type of mood, may be quite cryptic). I’d assume this would work in the new version too…
The dwarf ended up leaving the craft workshop they were in, and went to the craft workshop next door, after grabbing some bones from an animal I slaughtered. Viewing the workshop gave no indication of what had been claimed for use though. There were a couple of icons above the dwarfs head which I now think were to indicate they were working on the artifact. A red cross with green squiggly arrow perhaps indicates that the dwarf is constructing, and the icon that looked like an insect was perhaps indicating they were in a mood, or the specific mood.
All the changes will take some time to get used to, at least until the community has them figured out.
Edit: Just had a closer look at the dwarf and it was a Bard that I allowed into the fortress to entertain. Perhaps because they aren't a citizen moods work differently. I only noticed when I tried to add them to a military squad but they weren't on the list.
Further Edit: I had a mood from a regular citizen dwarf and it describes everything that is happening in the workshop icon. So my problem was just because the dwarf had petitioned to come to the fortress, and not been born/migrated there.