Look at the classic character Droopy Dog. Everything about him seems tired, slow, dejected, even a bit dull-witted. Look at the way he's drawn, built from the ground up to just be droopy all over. It's not just his face that sags; it's his entire body. His ears droop almost to the ground. His stomach is a ponderous swell sagging almost to his knees. His limbs are attenuated and dwarfed, while his head is extremely prominent, forcing attention to the accentuated sags and bags. When he moves, it's not just his face that sways sluggishly. It's his entire body, letting us know that all of him is sagging and tired, not just his face.
You can add these little nuances to your characters, as well. When a character recoils in fear, you can make their ears stand up, stretching out beyond normal proportions if need be. Their bodies might go stiff, stretching out and shaking - a visual example of the expression "scared stiff." It's much more effective than having them stand at a casual "at rest" posture with only their faces registering fear, shock, or horror.
The shoulders can play a large part in animating character expressiveness, especially when combined with the hands. Imagine your character is telling another that they don't care. Their face might be bland, expressionless, or even condescending and dismissive. But what really says they don't care is the shrug of one or both shoulders, paired with that little casual flick of the hand brushing off whatever the other character is concerned about. In another situation, your character may be exhausted. Their limply open mouth and half-lidded eyes may convey that, but it's brought home by shoulders that sag so heavily they appear narrower, dragged down by the weight of hands hanging limp at the sides, nearly draping to the floor. Those hands may be so heavy they bow the entire body forward, implying the exhaustion is both dragging them down below and pushing down on them from above. When hunched forward, the shoulders can convey anything from fear to anger, depending on the posture.
Talking with the hands is another way to convey emotion, especially excitement or interest, though it works all across the spectrum. If your character is talking about an enormous fish they'd caught, they wouldn't just say "I caught a fish, and it was pretty big." They'd say "I caught a fish, and it was thiiiiiiiiiiiis big!" while stretching their arms apart to convey the size of the fish. Conversely, they might be reluctant to talk about how small the fish they caught was, and that shame would show in how they hesitantly pinch their fingers together. Clenched fists can convey aggression as much as clenched teeth. Hands tucked behind the back show shyness as much as a downcast head.
Emotion is a complex and subtle thing, and learning to portray it properly is about more than just watching the face and mimicking what it does. Study body language. Watch the people around you; just try not to be creepy about it. Learn how people show emotion with more than just words and facial expressions, and learn to infuse that into your animations. Suddenly your characters won't seem wooden anymore.
