Chere Madame Bea,
Please forgive the length of this post, and my poor attempt in using Shakespeare���s language, Moliere is more my cup of tea.
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Your French teacher is absolutely right using Adieu is very old fashion. So are Alex and Musketeer when they wrote that Adieu is almost never use, in vernacular speaking. On the stage, in the movies, or as we say in French pour faire de l���esprit (to be witty) Adieu is still understood and used. But in regular daily life it is not at all used. The usage of a specific word have to be understood in it context.
Language is an abstract concept; it is not a fashion show! I am old fashion and I am rather proud of that. You might even find theses days some people thinking of G*d as an old fashion concept.
Even thought English is the clearest of language (that is why, in my opinion, is it so powerful) French have more nuances that are easily misunderstood by foreigners.
It is very a propos that this question was asked this week. This is the only week during which I find myself using Adieu a few times. When the graduates students leave university and enter the working world, I put some time aside to wishes them farewell. With most of them, we finished the conversation with Au revoir. For the exceptionally good one, the ones that have the greatest hopes I finished with Au revoir but my heart is recommending them to Him so I think Adieu. Hope this help.