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I am currently editing an original work of fiction, and one of the things I'm having the most trouble with is dialogue and paragraphs after certain character's actions. I know it sounds a little confusing, but I'll try my best to explain.

In this example paragraph, there are three characters: Nat (the narrator), Oddie, and Arden.

Would it be written:

“We can just call her Rose,” Arden suggests, leaning back on his
hands, any leftover semblances of sleep wiped clear from his face.

“I thought you said that name was already taken?”

He shrugs, toeing the remnants of his cloak over his lap. I throw my
arms in the air, exasperated.

or,

“We can just call her Rose,” Arden suggests, leaning back on his
hands, any leftover semblances of sleep wiped clear from his face.

“I thought you said that name was already taken?” He shrugs, toeing
the remnants of his cloak over his lap. I throw my arms in the air,
exasperated.

Basically, I'm wondering if I should start a new paragraph after the dialogue if the action is being taken by a person other than the one who just finished talking.

I know it sounds a bit confusing. I'm very sorry if I don't explain myself well. Still, I hope someone will be able to help me, as I'm having some trouble with this. The second option seems right to me, but in writing it like that, I'm afraid the reader might get confused about who's doing what.

Any help will be greatly appreciated!

I would say it is definitely a new paragraph if only to indicate 'he' didn't say 'I thought ...'.

I teach that you start a new paragraph when you change speaker, place, time or character. Here the change is character.

First of all, I'll admit I had some trouble identifying who said

“I thought you said that name was already taken?”

I'm assuming that Oddie said it in reaction to Arden's suggestion and that Nat, the narrator, got exasperated when Arden shrugged. Please let me know if I got it wrong.

Option 2

“We can just call her Rose,” Arden suggests, leaning back on his hands, any leftover semblances of sleep wiped clear from his face.

“I thought you said that name was already taken?” He shrugs, toeing the remnants of his cloak over his lap. I throw my arms in the air, exasperated.

This makes it look as if the speaker shrugs while saying the line.

Option 1

“We can just call her Rose,” Arden suggests, leaning back on his hands, any leftover semblances of sleep wiped clear from his face.

“I thought you said that name was already taken?”

He shrugs, toeing the remnants of his cloak over his lap. I throw my arms in the air, exasperated.

This does not give the impression the speaker of the line and the person who shrugs are the same.

Note:

If you feel the reader might get confused about who is saying what, use the character's name or an attribute that clearly identifies the character.

“I thought you said that name was already taken?”

Arden shrugs, toeing the remnants of his cloak over his lap.

Imagine that Arden's cloak is blue while Oddie's brown.

“I thought you said that name was already taken?”

He shrugged, toeing the remnants of his blue cloak over his lap.

Yes. Not only should you start a new paragraph for every character, but you need to be clear about who is saying or doing what.

You mention the excerpt involves 3 characters. We know the first person part is Nat, the narrator. And of course the parts you label as being from Arden are from that character. But where is Oddie? Is he the speaker in the middle? If so, try this:

“We can just call her Rose,” Arden suggests, leaning back on his
hands, any leftover semblances of sleep wiped clear from his face.

Oddie looks up. “I thought you said that name was already taken?”

Arden shrugs, toeing the remnants of his cloak over his lap.

I throw my arms in the air, exasperated.

It's considered normal to start a new paragraph when somebody new speaks; however, it's not essential. I've read more than a few books where dialogue from two people happens in the same paragraph. When done correctly, it can be done without any confusion.

The main point, however, is that it needs to be made clear who's speaking.

Switching to a new paragraph is a stylistic way of indicating that the speaker has changed. But just switching paragraphs may not be enough. For example, there could be more than two characters. Switching to a new paragraph will likely not be sufficient to indicate that the conversation has switched out of an alternating discourse between two people and it's now a third or fourth character interjecting themselves into the dialogue.

What can also be disconcerting is when the same character is speaking in a new paragraph. This is normally indicated with the lack of a closing quotation mark at the end of the previous paragraph. Nonetheless, that may not be easily spotted—and if it's missed, it can cause confusion.

Whether it's different people in the same paragraph, different people in alternating paragraphs, or the same person in multiple paragraphs, narrative markers can be very helpful. If it's not clear (no matter what the format), break the speech to identify the speaker in some way.

Incidentally, regardless of the paragraph breaks, it's not clear who is speaking the second line of dialogue in the passage you've provided.

It's likely Oddie—but it could also be Nat. (It could even have been Arden, continuing on from the previous paragraph, if different words had been spoken.)

Unless the speaker is clear when viewing the passage from its larger context, I would say you should give some kind of explicit indication of who that second line of dialogue is coming from.

I've seen both forms used, but in the first one it feels like he says it, and then he shrugs. While in the second one it feels like he is saying it as he is shrugging. Hope that helped <3