I agree with the sentiment, stop doubling down; or tripling down:
The prince looked abashed. “I- I thought I was being polite,” he mumbled.
You have three indicators of the same emotion in this line; abashed, a (presumably non-typical) verbal restart, and mumbling.
If this were my own writing (and it could be, I also have this problem) I'd ask myself, how else might we intensify this emotion? Shall he hide his face in his hands? Can we have him avert his eyes, too? While bowing his head, and crossing his arms in unconscious self-protection? Kick a little dust to the side with the toe of his boot? :-) (Actually not a bad exercise, if you pick only one.)
Technically speaking, "he mumbled" is not necessary, since it is clear he is speaking after "The prince looked abashed". These are single-indicator versions.
The prince looked abashed. "I thought I was being polite."
"I -- I thought I was being polite," the prince said.
"I thought I was being polite," he mumbled.
Now I admit each, when read, evokes a slightly different image of the prince and the sound of his statement. but that is okay, choose the one you like. The point is to avoid trying to intensify every image with 2, 3, or 6 indicators of emotion or mood, except perhaps in a particularly fraught emotional moment when intensity is warranted.
But your main question is how to not always use the same form of description; how somebody "looks".
For a POV character, I might just state it.
The prince was abashed. "I thought I was being polite."
Or, I get in his head,
I've made her think me a fool!
"I thought I was being polite," he said.
Or, a visual clue.
The prince closed his eyes for a moment, then met the eyes of the captain. "I thought I was being polite."
Trust your reader. You don't need an iron fist on the exact emotion. It slows down the reading to double and triple, without actually paying off in a brighter image. Your reader is working to imagine the scene, they already know the relationship between these speakers (which limits the number of ways the prince might feel here; e.g. it might rule out anger or disdain).
If you like that pace, use the extra words to actually paint a brighter image, instead of saying effectively the same thing two or three ways.
When dealing with non-POV characters, you can still filter images of how somebody looks through the POV's thoughts.
I've embarrassed her. God above! Shall I ever master conversation with a lady?
Not only does this clue the reader in to how his conversational partner looks, but we get a sense of how the prince feels and must look as well.
Instead of "the prince looked confused,"
Didn't see me? What riddle is this?
"I mean," the captain said, "You seem to be acting a part, and poorly. Did you believe your compliments, or cast dice to choose them from a list?"
So, with the "I mean," (or "What I intend to say," if the captain wants to be more formal addressing a prince) we know the captain has interpreted whatever expression went with the prince's thoughts, and is continuing and clarifying what he meant. We do have to attribute the dialogue to the captain, since he speaks two lines in a row, but we don't have to say "he continued," that is obvious from the fact that he is speaking again.
Without saying what the prince looked like, readers typically assume expressions will follow thoughts, and see what the captain sees, confusion. But being in the thoughts of the prince feels more intimate.
If you are writing in something besides 3rd Person Limited, you can still use an indirection instead of "telling" that the prince is confused.
The captain realized the prince was trying to riddle out his meaning. "In other words, prince, you seem to be acting a part, and poorly. Did you believe your compliments, or cast dice to choose them from a list?"
"[Character] looks [emotion]" is telling. It is appropriate for a screenplay, the actor will interpret this into "showing". And it isn't always inappropriate in a novel, but you are right, it can be overused.
The trick is often to not state it, but have the emotion cause something else, particularly thoughts or reactions in other characters. Let the reader infer the emotion. Because you can also overdue physical responses, averted eyes, frowns, and so on, and there is a limited menu of those before they become repetitive or comical.
But there is an unlimited number of thoughts and reactions that can be specific to the situation. So the prince can be confused thirty times in the book, but in each instance his confusion is about some paradox in his mind that is different and specific to each situation, so you never say he is confused. And you don't need some physical clue, him putting a hand to his temple every time he is confused.
Now I'm not banning either of those, I'm just saying they are unnecessary, because every confusion is different, and the question in his mind is different, so the reader gets that he is confused and why each time. And the same for every "angry", "bored", or other emotion.
Getting into his thoughts also has the advantage of letting him be non-expressive when you wish. His thoughts can still show the reader he is an emotional person, but he doesn't have to wear every emotion on his face for other characters in the story (one of the few advantages of a novel over a screenplay).