How do I make make objects stick to a ball and affect (but not stop) its movement (like Katamari Damacy)

I'm pretty new to Unity. I've done a few tutorials now and I'm pretty familiar with the editor and have a very basic understanding of scripting.

I've done two different tutorials now that have involved rolling balls and I have not been able to get Katamari Damacy off of my mind.

Thanks to some helpful advice from Nesis and some experimenting, the pickup objects (cubes) now stick to the ball. However, as soon as a cube sticks to the ball, I lose pretty much all control over the ball. Sometimes I can still move it a bit but I have very little control over which way it goes. What I want to happen is that I can continue to roll the ball but the cubes throw off the rolling with their shape causing it to wobble. If you're having trouble understanding what I mean by this, just go watch a gameplay video of Katamari Damacy on youtube and you'll see what I mean.

Currently, I am achieving this "sticking" effect with a script attached to the pickup object (cubes). Both the cubes and the player objects (the ball) have two colliders, a non-trigger which is the same size as the object (to make the objects solid), and a trigger collider that is .1 units larger than the object mesh.

I've tried several different things so far, such toggling gravity, kinemetics, and isTrigger on collision but none of that helped.

Edit: I just noticed that if I roll the ball into so that it gets stuck to an outer edge of the ball (from the player's perspective behind the ball) then the ball will keep rolling normally, but the cube just clips through the floor, it doesn't affect the rolling in any way. However, if you roll over a cube, you roll over it (not through it) and it sticks just as it should, but when the ball rolls around to where the cube is on top the ball stops rolling and sometimes spins like a top. :-/

Here is the script for objects that can be picked up:

 using UnityEngine;
 using System.Collections;
 
 public class Sticky : MonoBehaviour 
 {
     public bool isSticky;
     private bool isStuck;
     
     void OnTriggerEnter(Collider other)
     {
         if ((other.gameObject.CompareTag("Player")) && (isSticky == true) && (isStuck == false))
         {
             transform.parent = other.transform;
             isStuck = true;
         }
     }
 }

Here is the player control script (including this in case there is some change needed to player control in order for this to work properly):

 using UnityEngine;
 using System.Collections;
 
 public class PlayerController : MonoBehaviour 
 {
     public float speed;
 
     void FixedUpdate ()
     {
         float moveHorizontal = Input.GetAxis ("Horizontal");
         float moveVertical = Input.GetAxis ("Vertical");
 
         Vector3 movement = new Vector3 (moveHorizontal, 0.0f, moveVertical);
 
         rigidbody.AddForce(movement * speed * Time.deltaTime);
     }
 }