Alright, so here's the first thing. When writing a story, it's so easy to fall into the trap of considering the action the 'real plot' and the emotions 'not the real plot' don't do that to yourself, because it can easily cause half of your problem. The emotions of the characters are definitely as crucial as the action itself especially if, as you said, the villain can only be stopped by emotional confrontation. So in that context, you have to make sure that you give emotional moments the proper amount of time. Plus, sometimes action scenes need a moment to breath to give them the proper amount of significance. going from one to the other might make them lose some of the impact because it feels so constant.
My suggestion, don't go straight from action scene to action scene. Rather, give the plot some transition time, and that's where you include the emotions. Don't, obviously, just add a lot of filler, but you can give them moments to breathe, possibly, have a mental breakdown directly after a battle, and be almost unable to function and the other characters have to try to bolster them up a bit. Then their conversation gets interrupted suddenly and it's back to action.
Example: Say they have to go storm the villain's lair, and they are riding the train to do so. Leader of the rebel group is going from person to person to make sure they all are in the proper condition, armed, wearing protective gear, and they notice that one member definitely seems to be having a hard time. They ask them what's the matter, and the person expresses how they can't understand how everyone can be moving on like this when this is so insane. They all almost died, they were this close to losing (insert character) how can everyone else be so composed and stoic. Leader explains something to them, say...that everyone they love will be lost if they don't keep their heads about them and deal with the threat. The person struggling with the emotional meltdown tries to say something else, but just then they are stormed by the enemy who sees them coming, or the lights on the train begin flashing to let them know it's 'go time' 'now, or never.' something along those lines.
For my novel "Damsel in the Red Dress," there are few really climactic scenes of life or death or really traumatic intense experiences, but I lead up to them with scenes that put them into the emotional state I need them to be in to have the proper impact in the story.
It would be really long to post those here, so if you'd like to see my examples, I can link the chapters for you. Either way, I hope I helped you.
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Again, I know this is a lot, and you certainly don't have to read these, but it might help you figure out at least somewhat, how to do what you're trying to figure out. This is a romance, rather than an action story, which I think is what you're writing, but when it comes to writing emotion, it might actually be really helpful, since romances tend to put more focus and emphasis on emotions than action series do.