Trying to get a basic first person camera working only one axis rotating

my code is parented to a camera in unity 3d, for some reason, it is only letting me move on one axis at a time. When I comment out a Quatornian.AngleAxis line suddenly the other line will start working and I can smoothly rotate the camera with the mouse on that axis.

  public class CameraMovement : MonoBehaviour
    {
        [SerializeField] private float multiplier = 100.0f;
        private Transform cameraTransformControls;
    
        [SerializeField]
        private float xRotation, yRotation = 0;
        // Start is called before the first frame update
        void Start()
        {
            Cursor.lockState = CursorLockMode.Locked;
            cameraTransformControls = this.GetComponent<Transform>();
        }
    
        // Update is called once per frame
        void Update()
        {
            //Camera.main.ScreenToViewportPoint(Input.mousePosition)
            //cameraTransformControls.Rotate(Input.GetAxis("Mouse Y") * multiplier, -Input.GetAxis("Mouse X") * multiplier, 0.0f, Space.World);
            float mouseX = Input.GetAxis("Mouse X") * multiplier * Time.deltaTime;
            float mouseY = Input.GetAxis("Mouse Y") * multiplier * Time.deltaTime;
    
            xRotation -= mouseY;
            xRotation = Mathf.Clamp(xRotation, -90f, 90f);
    
            yRotation -= mouseX;
    
            transform.localRotation = Quaternion.AngleAxis(xRotation, Vector3.right);
            transform.localRotation = Quaternion.AngleAxis(yRotation, Vector3.up);
        }

Let me correct your script :

float mouseX = Input.GetAxis("Mouse X") * multiplier * Time.deltaTime;
             float mouseY = Input.GetAxis("Mouse Y") * multiplier * Time.deltaTime;
     
             xRotation -= mouseY;
             xRotation = Mathf.Clamp(xRotation, -90f, 90f);
     
             yRotation -= mouseX;
     
             transform.localEulerAngles = xRotation * Vector3.right + yRotation * Vector3.up;