Making a puppet show like walk animation?

I should start off by saying I have no idea what I’m doing, so I suspect I’ll be coming here to ask random questions a lot. This is my first experience with game making. So, for my first game I wanted to make a 2d platformer inspired by retro 8bit grap- no, we have too many of those. Instead, a game where everything looks like it’s a puppet show and all the characters are held up by popsicle sticks. Think kinda like how Battle Block Theater does its cutscenes. So, they wouldn’t be animated exactly, but whenever the player moves in a direction, the character would bounce up and down while moving in that direction. I cobbled together a quick animation in After Effects so that you know what look I’m going for…

Its a spooky spooky ghooooosstttttttttt… Anyway, I don’t want to make the character’s walk animation literally be it moving up and down because if the player stops mid bounce, it shouldn’t snap back to the middle. That would look terrible. Plus, the stick that’s “holding” it up should also be bobbing up and down. And I really don’t want the camera to bounce as well because that would be nauseating. Is there a way to make the character sprite bounce up and down independent of the hitbox then have the stick inherent that? And once the player stops, can it glide back to its normal position naturally? Or is there some other, easier way to do this?