I think I read this in the early 1990s. Possibly in Asimov's or Analog, but more likely in an anthology.
The viewpoint character is a man who has been increasingly insomniac. Restless, he has taken to walking through city streets late at night.
He sees a woman get out of a cab, walk to a club perhaps? She drops her shawl, as she goes in. He goes over, finds it dissolving in the gutter.
At some point he sees her again, and feels compelled to follow her. He stalks her home; when he enters a short time later, he discovers her and some others sitting, all frozen like mannequins. He flees.
At some point he realises that he is becoming like them; he is one. He is in a bar, having grown a jacket; he reaches into his "jacket" and a slit opens and extrudes some money (bills). I recall him noting as he touches them they are still faintly damp, but they dry quickly as he passes them to the bartender.
In the final lines he and the woman sit together on adjacent stools, and have sex with organs in the side of their hips.