That is an episode from the mid-eighties radio show *Midnight*. The title of that episode is indeed "My Cup Runneth Over".  The show is exactly as you remember it.  You can listen to it (and many other episodes of *Midnight*) [here](http://www.fearyoucanhear.com/shows/andromedacbsrmt.php?q=f&f=%2FMidnight).