You can try Weasyl, inkbunny, or pixiv.

I usually copy the story into the description, or upload it as a pdf, or both

As somebody that just sorta browses once in a while without something I am actively seeking, I find my attention is grabbed by custom thumbnails more than that blank "Story" default icon... but beyond that I have no idea what works best or what is "standard" here.

I get the feeling that if you posted a story, most eyes on it would be your existing followers so the thumbnail wouldn't be as relevant as an interesting title. I recall a story of yours that I read over on DA years ago, about Maurice being stuck in a theater at night or during a storm (or both? it's been a while since I read it I apologize) and what interested me in that was my familiarity with your Maurice character more than the bland wall-of-text preview thumbnail that was a literature preview on DA at the time.
If I have a story, it typically goes with a drawing of mine so I just put it in the description of said drawing. I'm better with ideas than narratives so that typically works for me, writing short stories to supplement a visual work rather than writing standalone stories.

That's my perspective as a simple viewer, and as someone that occasionally has what is effectively a tl;dr description for a drawing of a fat cartoon animal being awkward. However, I have no idea about any of the technical aspects of literature posting here, but I can ask around, as some of my friends seem to be writers.

I would recommend making your own thumbnail for it. I’ve also been uploading to WattPad.