How does your slime physics work? If you made the script for it, meaning you have control over the points and their velocities, then just have the jump method loop through the points and set their velocities up. If not, what library/components are you using?
Could you post a screenshot of the Inspector for the ball and if they are separate, for one of the spheres inside?
To what body do you apply the force? My hunch is that you will want to apply the force equally to all parts of the ball, not just the center object because this will otherwise drag the inner spheres behind and might cause them to spin.
Also on the joint, “enable collision” should either be off or the setup needs to leave a little space between each sphere collider, they must not touch or even intersect.
Yes. I suppose the wobble is mainly due to accelerating the main (center) body. This forces the spring joints to follow, thus adding several more forces on the slime which, when they interact, cause the slime to wobble to some degree.
Try applying the same jump force to each Rigidbody on Player and its children so they all move uniformly in the same direction.