Third Person Camera Roatates Diaginal

Hello Everybody, I am trying to make a third person shooter and am trying to make the camera rotate around an object with the direction of the mouse. I used the script below however I don't even think I should for a third person shooter, but the real problem is when I start to move my mouse, the camera starts to move diagonal, so I tried to freeze the Z axis through a rigidbody although I knew it would change as the camera orbits my player/object, though it still rotated. So I was wondering if there is anyway around this or if there is another way to approach a third person shooter although I haven't found any which is why I tried this

 using UnityEngine;
 using System.Collections;
 
 public class CameraScript : MonoBehaviour
 {
 
     public GameObject target = null;
     public bool orbitY = false;
     public int lookSpeed = 30;
 
     void Start ()
     {
     
     }
     
     void Update ()
     {
 
         if (Input.GetAxis("Mouse X") < 0)
         {
             transform.RotateAround(target.transform.position, Vector3.up, Time.deltaTime * lookSpeed);
         }
 
         if (Input.GetAxis("Mouse X") > 0)
         {
             transform.RotateAround(target.transform.position, Vector3.down, Time.deltaTime * lookSpeed);
         }
 
         if (Input.GetAxis("Mouse Y") < 0)
         {
             transform.RotateAround(target.transform.position, Vector3.right, Time.deltaTime * lookSpeed);
         }
 
         if (Input.GetAxis("Mouse Y") > 0)
         {
             transform.RotateAround(target.transform.position, Vector3.left, Time.deltaTime * lookSpeed);
         }
     }
 }

Thank you all for your time :)