So... is Houkago no Shokuinshitsu finished or not?
NO, it is not finished.
Why?
The anime finished after the manga:
Anime finished: January 27, 1995
Manga finished: January 01, 1992
... But MangaUpdates contains information about the anime adaptation and indicates that it is incomplete.
Watching Order (by release date)
Here you have all the anime relations we detected ordered by release date, so you can see the reason for this status in context and also use it as watching order.
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Synopsis
Houkago no Shokuinshitsu follows Kawase Mitsuro, a newly hired art teacher at a middle school. He's immediately aware of the attention senior teacher Kazama Toshiaki receives from the female students.
However, Kazama seems to take a particular interest in Kawase, inviting him out for dinner.
Kawase is skeptical of Kazama's intentions, but agrees to the outing.
The story explores the budding relationship between the two teachers as Kazama's genuine feelings come to light, and Kawase grapples with his own desires and the societal pressures he might face.
The central conflict revolves around whether Kawase will choose to pursue a relationship with Kazama or succumb to external expectations.