Do you see Hailey move or it just the camera that shifts?
You can try destroying the first projectile and then jump on the platform or time your attack perfectly to destroy it's projectile and hit it when it's close to the edge.
Viewing post in (WIP) Hailey's Treasure Adventure comments
I've found that the only way to kill the red Skelton is to destroy the first projectile and hope it walks close enough for you to hit it before it sends another
As for the walking, I've only had it happen twice. When the animation finishes she gets up and immediately starts walking to the right as if you had the button pressed and doesn't stop until you press it the button to walk right yourself