The viewpoint character's emotions and senses must be as much at play in a fight or action scene as his body and weapons.
I make a special effort to include all the senses in my descriptions. What does he hear? See? Smell? Taste? Feel?
How do he react to killing someone? The death of a friend?
Adding emotion isn't an either/or situation. It's just as vital to add emotional layers to the physical action as it is to have brief moments of introspection when the battle isn't going on.
Characterization also isn't just introspection. It's characters interacting with each other and revealing themselves in bits and pieces.
Your band of adventurers may not sit around "sharing their feelings" in touchie-feelie moments like a Dr. Phil show, but they've been around each other enough to know that one hates the bad guys because they murdered his wife and kids, and he's liable to attack without thought and ruin their surprise attack.
He may be clutching the sword at his side, his other hand opening and closing in nervous energy, and another adventurer may warn him to relax and may mention the wife and kiddies.
The image of his wife's raped and brutalized body could flash through his mind, and he fights his raw anger and lust to kill. That won't slow the action down like having a long interior flashback of him finding his family's bodies, and his vow of revenge. Instead, it adds to the excitement of the coming action because the reader now questions whether this guy will lose his cool and get everyone killed.
An even better way to present this information is to put it in an earlier scene that isn't action intensive so the reader will know the details and will only need a slight reminder of this character's motivation and tendency to attack without thought.
Remember, though, that a character's emotions are meant to increase the intensity of the scene, not slow it down or mar its pacing.