Dragging a 2D object that has a hinge or spring joint

This is pretty simple, I suppose. I have a cat object made of several sprites (arms, legs, head etc). They are all sprites with a Rigidbody and Poly collider. They are all attached together with Hinge joints (I have another with Spring but doesn't work as well).

I attach a Mouse/Touch drag script to the parent Cat object. It works fine. I can click and move the cat around. But as soon as I do so, it does not maintain the distances defined for my hinge joints (which are as small as allowed). I end up with a cat split into many pieces (oh god!) until I unclick and he flies back together. This is the script I am using that I found for Drag.

 using UnityEngine;
 using System.Collections;
 
 [RequireComponent(typeof(PolygonCollider2D))]
 
 public class Drag : MonoBehaviour {
     private Vector3 screenPoint;
     private Vector3 offset;
     
     void OnMouseDown() {
         
         offset = gameObject.transform.position - Camera.main.ScreenToWorldPoint(new Vector3(Input.mousePosition.x, Input.mousePosition.y, screenPoint.z));
     }
     
     void OnMouseDrag()
     {
         Vector3 curScreenPoint = new Vector3(Input.mousePosition.x, Input.mousePosition.y, screenPoint.z);
         Vector3 curPosition = Camera.main.ScreenToWorldPoint(curScreenPoint) + offset;
         transform.position = curPosition;
     }
 }

I'm just wondering how to make this more like a ragdoll. I cant seem to get the hinge joints to maintain their distance while I drag one of the sprites.