I am trying to find a short film which starred Judge Reinhold as a
college professor and Stockard Channing as another professor with whom
he starts an improbable affair.  I saw this film in Greece on Channel
7X sometime in the mid-nineties, when the fledgling station was
notorious for showing unreleased and commercially unsuccessful
material.  The showing ended with a banner titled "Trying Times"
(which could be the name of the film), and the credits soundtrack is
"It's a Sin to Tell a Lie" recorded, I believe, by Ruth Etting.

If you can find the name of the film, the year of release, and
somewhere I could purchase a video or DVD, that would be fantastic.

Well, it doesn't seem to be in the Internet Movie Database.  "Judge
Reinhold has a Stockard Channing number of 2" and vice versa according
to this "Six Degrees of Kevin Bacon"-like site:

"The Oracle says: Judge Reinhold has a Stockard Channing number of 2"
The Oracle of Bacon at Virginia
http://www.cs.virginia.edu/cgi-bin/oracle/movielinks?firstname=Stockard+Channing&secondname=Judge+Reinhold&using=1&game=0

"The Oracle says: Stockard Channing has a Judge Reinhold number of 2."
The Oracle of Bacon at Virginia
http://www.cs.virginia.edu/cgi-bin/oracle/movielinks?firstname=Judge+Reinhold&secondname=Stockard+Channing&using=1&game=0

[You can repeat these searches with TV or Movies & TV and get the same
basic outcomes.]