To answer the first part of your question (were there terms that could cause offense that people were supposed to not use): Yes.

To answer the second part of your question (was there a widespread movement to avoid using these terms, and to be aware of their harm): Kind of yes, but with some qualifications.

There is plenty of evidence of speech being considered insulting or demeaning, especially in politics, or relating to one's sexual practices, financial state, heritage, nationality, etc.

Some examples include:

Calling someone a "king" in the Roman Republic (Romans famously despised monarchies, and accusing a political rival of seeking a crown could lead to death). Romans also would accuse rivals of effeminacy, womanly behavior, or submissive seuxal practices.

In Norse culture, there were the Grey Goose Laws

There are three words—should exchanges between people ever reach such dire limits—which all have full outlawry as the penalty; if a man calls another ragr, stroðinn or sorðinn. As they are to be prosecuted like other fullréttisorð and, what is more, a man has the right to kill in retaliation for these three words. He has the right to kill in retaliation on their account over the same period as he has the right to kill on account of women, in both cases up the next General Assembly. The man who utters these words falls with forfeit immunity at the hands of anyone who accompanies the man about whom they were uttered to the place of their encounter.^(1)

In many cultures, people understood what words would cause offense, and did their best to avoid those words... unless they deliberately wanted to cause offense. The Romans called these invectives, and they could be brutal. At times, they would directly threaten to rape an opponent on the floor of the Senate house.^(2) Although these would often be brushed off, serious ones could lead to brawls, mob violence, or even assassinations. Julius Caesar didn't get stabbed because everybody liked what he was saying. Less dramatically, going too far or insulting the wrong person might cause you to lose political allies, popular support, etc.

This ties into the idea of dueling in a number of societies. As mentioned about, Norse culture had the Holmgang, a properly handled legal fight to settle various disputes, including insults. You can also see this in many other cultures: Japanese samurai, medieval knights, and throughout Europe and the Americas in the 1700s and 1800s. In most cases, certain words or aspersions on someone's honor were widely viewed as a valid reason to challenge that person to a deadly fight. In many of those societies, people were then trained to be careful and respectful to avoid giving offense (hence so many of the traits associated with being a "gentleman" in England).^(3) There's also the Code Duello, an Irish set of rules which then spread throughout America. It lists what kinds of offenses determine what response, how a duel should be organized, etc.^(4)

So to sum up the answer to your question: Yes, people recognized certain terms or insults as being harmful and cruel. Although their methods tended to be more violent than our own, people were motivated to avoid giving offense, and not use those terms lightly. However, many of those societies would then be fine with people using the terms, so long as they could back it up, often through a fight. So the motivation was less about kindness and respect towards others, and more about picking your moment to say something horrific.

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Sources:

1: Sørenson, Preben M.; Turville-Petre, Joan (transl.) (1983). The Unmanly Man: Concepts of Sexual Defamation in Early Northern Society.

2: The Caesars by Suetonius, specifically "The Divine Julius". Also, Roman Manliness: "Virtus" and the Roman Republic by McDonnell has some great stuff on the linguistics side of this, which may fit with what you're looking for. Finally, Making Men: Sophists and Self Presentation In Ancient Rome by Gleason has some great content on the exact rigorous standards Roman men had to keep during oratory.

3: Etiquette for Gentlemen, being a Manual of Minor Social Ethics and Customary Observances (1857, author unknown).

4: Code Duello