There are a few connections between Poe's works, generally via one story being directly or indirectly referenced in another one published later. However, there are two instances of multiple stories having the same universe, and sometimes the same characters: A Predicament and How to Write a Blackwood Article, and The Landscape-Garden, The Domain of Arnheim and Landor's Cottage.
1. Ligeia and The Conqueror Worm
In 1838, Poe published Ligeia, a short story about a man struggling to come to terms with the illness and then death of his first wife (Ligeia), and then the death of his second wife in the same manner.
Five years later, Poe published The Conqueror Worm, an extremely dark poem focusing on the inevitability of death, likely largely drawn from his own experiences with mortality - as was the case with much of his works; Poe drew quite a lot of inspiration from the lives and deaths of those he loved. Two years later, in a republication of Ligeia, he added The Conqueror Worm to it - written, or rather spoken in a fit of madness, by Ligeia herself. Shortly after finishing speaking the poem, she dies, after a brief addendum.
The two works obviously do not take place in the same universe; one is simply a part of another. However, I feel that such a connection is worth including. This was the only other explicit connection I could find between Poe's works (though the characteristics of Dupin are not limited to that character alone). The theme of humans facing the horror of death does permeate much of Poe's writing, and it is perhaps a large part of his legacy.
As for why Poe included the poem in the revised version of Ligeia - well, there are many interpretations. Wikipedia states
Because it emphasizes the finality of death, it calls to question Ligeia's resurrection in the story. Also, the inclusion of the bitter poem may have been meant to be ironic or a parody of the convention at the time, both in literature and in life.
This is certainly a valid interpretation - and one I agree with, because, to be frank, Ligeia is really about whether or not death can be overcome. Granted, some feel that Ligeia did in fact conquer death, but the same basic reasoning is used: Poe included the poem to just emphasize the human struggle with mortality.
As an interesting side note, the name "Ligeia" also appears in Poe's poem Al Aaraaf (1829).
2. A Predicament and How to Write a Blackwood Article
You brought up a pair of short stories, published together under different names in 1838. Both have the same protagonist, who refers to herself as the Signora Psyche Zenobia. The plots are otherwise remarkably bizarre and separate from one another; the former is about the odd decapitation of said protagonist, while the latter is something of a social commentary on horror and suspense writing, which was of course Poe's specialty.
If I'm interpreting them correctly, A Predicament is the story of how the Signora Psyche Zenobia inadvertently (I believe) carries out her assignment from How to Write a Blackwood Article: to die and record her thoughts on the matter. The details I leave to anyone who wishes to read the stories in full.
3. Poems written as "Tamerlane"
This is another peculiarly connected group of poems. You may know of Poe's 1827 poem Tamerlane, published in his collection entitled Tamerlane and Other Poems (1827). What's curious is that Poe actually signed some of his poems as "Tamerlane", for reasons I cannot explain. They include
- To -- (also known as "Sleep on"), published in 1833 in the Baltimore Saturday Visiter.
- Fanny, published in the same periodical a week later.
I have been unable to figure out if there is any reason behind Poe's choice of pseudonym, or if he intended these poems to be additions to Tamerlane and Other Poems. They are not strictly stories, but I felt it might be interesting to include them, given that Tamerlane itself is a long enough epic poem to be on the same level as a story. It is possible that Poe intended the connection to be similar to that between Ligeia and The Conqueror Worm.
4. The Landscape-Garden, The Domain of Arnheim and Landor's Cottage
In 1842, Poe published The Landscape-Garden, which he subsequently extended greatly into The Domain of Arnheim (1846-1847). The original short story centers around a (very wealthy) young man named Ellison, who wants to shape the landscape like an artist, with the country his canvas. The newer version actually explains how Ellison goes on to create Arnheim, a castle that fulfills his vision.
Finally, we have a tenuous connection with Landor's Cottage: A Pendant to The Domain of Arnheim. Though neither Ellison nor Arnheim itself are mentioned by name in this story, it becomes obvious that this story is set in the same universe as the previous two - just later in time. The unnamed narrator is walking through rural New York State, and comes across what remains of Ellison's creation.
I think this may finally be what you were looking for: multiple stories set in the same universe.