What exactly is your process for developing animations? Do you keyframe the movement? Do you use any kind of simple skeleton? Or do you work step by step through the motion from one frame to the next?
What it appears to be is the latter, as this often leads to very reluctant movements in the animation, with only small parts of the sprite being modified. You need to keyframe the entire figure moving through the motion from strongest stance to strongest stance, focusing on the shifting of weight/support through the figure (and in this case also the sword).
You can keyframe with a simple silhouette (can help with volumes), a very rough block in (I find this better to do after one of the other two), or a skeleton (helps maintain proportion and mechanics of the figure in motion).
Try thinking of the motion in terms of the key structural points on the figure: the feet, pelvis, spine and head. How does the pelvis move over the space that the feet bound? Is the head above the pelvis or not? How does this affect the spine? Basically, "is the figure in balance, or not?" If they are not, they will need to come to a point of balance by the end of the animation.
But to iterate, do not do this process frame by frame, keyframe it from major stance to major stance, then tween each of those, then tween those, etc until you are happy with the level of smoothness you have.