Do Adenim (Jews from the Yemenite city of Aden) have a nusach and a siddur unique to them?
I finally found a Siddur that fits the bill! It is called Siddur Sim Shalom and is Nusach Aden. It can be found here: https://tablet.otzar.org/he/book/book.php?book=196528&width=0&scroll=0&udid=0
In my opinion, the Adeni nusach is probably most similar to Shamai. In some Adeni shuls they use books similar to the Zechor LeAvraham siddur (Shamai), but many now use their own Sim Shalom siddur (Adeni). To be honest, a lot of Adenim just use Edot HaMizrah siddurim because that's what they're used to.
There was an Adeni siddur published a little before Sim Shalom, but the contents are very similar.
A lot of the Adeni prayers do differ to both Shamai and Baladi. Some of the differences seem to be unique to Aden, others have been taken from Sephardi or Ashkenazi liturgy. Even within Adeni shuls there are differences in the prayer. As a point of history, the Jews of Aden (and in fact many communities) got their siddurim from Livorno and would just make do with any differences in the liturgy.