Drag a sprite around with mouse, but have it not overlap other sprites

Basically, the user can drag sprites around in the scene, yet they cannot overlap. I sort of managed to achieve this using colliders, however, I experience collider tunneling. It's a real headache - how to have the sprites (irregular shape, no primitives available) move around, but not overlap other sprites.

Edit: Alright, asking questions after 18+ hours of frustrating work is not a good idea! To elaborate more on the question - all of the sprites have got convex mesh colliders. I'm using rigidbody.movePosition to drag the sprite. This works really well if I move the mouse slowly, but whenever I start moving fast - the sprite, with its' collider, starts jumping right through other sprites and so forth. My first idea was to move the sprite with forces, setting a maximum speed, but this didn't quite work with my drag, because it would neither move fast enough, nor position properly when I let go.

Also, while this may seem strange - I'm working in 3D space, rather than 2D, because of the nature of the game and my own camera script, which is my pride and joy, but only works with a perspective camera.

In addition, I'm adding the main script I use for moving around.

 using UnityEngine;
 using System.Collections;
  
 [RequireComponent(typeof(Rigidbody))]
 public class Dragable : MonoBehaviour
 {
  
     public int normalCollisionCount = 0;
     public float collisionMoveFactor = .01f;
     public bool freezeRotationOnDrag = true;
     public Camera cam  ;
     private Rigidbody myRigidbody ;
     private Transform myTransform  ;
     private bool canMove = false;
     private bool gravitySetting ;
     private bool freezeRotationSetting ;
     private int collisionCount = 0;
     private Transform camTransform;
     
     void Start () 
     {
         myRigidbody = rigidbody;
         myTransform = transform;
         cam = Camera.main;
     }
  
     void OnMouseDown () 
     {
         canMove = true;
         gravitySetting = myRigidbody.useGravity;
         freezeRotationSetting = myRigidbody.freezeRotation;
         myRigidbody.useGravity = false;
         myRigidbody.freezeRotation = freezeRotationOnDrag;
     }
  
     void OnMouseUp () 
     {
         canMove = false;
         myRigidbody.useGravity = gravitySetting;
         myRigidbody.freezeRotation = freezeRotationSetting;
         if (!myRigidbody.useGravity) 
         {
             Vector3 pos = myTransform.position;
             myTransform.position = pos;
         }
     }
  
     void OnCollisionEnter () 
     {
         collisionCount++;
     }
  
     void OnCollisionExit () 
     {
         collisionCount--;
     }
  
     void FixedUpdate () 
     {
         if (!canMove)
         {
             return;
         }
  
         myRigidbody.velocity = Vector3.zero;
         myRigidbody.angularVelocity = Vector3.zero;
         
         Vector3 pos = myTransform.position;
         myTransform.position = pos;
  
         Vector3 mousePos = Input.mousePosition;
         Vector3 move = cam.ScreenToWorldPoint(new Vector3(mousePos.x, mousePos.y, 9.35f)) - myTransform.position;
         if (collisionCount > normalCollisionCount)        
         {
             move = move.normalized*collisionMoveFactor;
         }
         
             myRigidbody.MovePosition(myRigidbody.position + move);
     }
 }