I've been looking for this for ages. It was out of a sci-fi short story anthology I read probably about eight years ago, but it could have been published anytime between 1970 and then, since it was out of a library.
What I remember was a genderless shapeshifter (used a nonbinary pronoun like Xe or Ze, but I don't remember exactly what) that needed to ingest the DNA of something to shapeshift into it. A giant naked ogre kept showing up and killing the residents, so the government (?) asked the shapeshifter to use a particularly powerful creature's DNA to kill the ogre.
The creature could teleport across dimensions and dragged the ogre through a bunch of worlds and killed one of the really, really inbred rulers. In the end, it turns out that the ogre was a man dreaming about being an ogre and killing people. The shapeshifter puked up the DNA because the government laced it with stuff to make them puke, so they couldn't keep that powerful of a form.
The main character also had a companion that helped them, and liked sleeping on piles of furs and cloths because they liked the way the different textures felt. I know the author had a website with mention of wanting to make more stories.