How do I stop my character from moving up and down while walking?

I've figured out that the problems with transform.forward but am looking for a way to move in the direction the characters facing without it trying to fly off, as it messes with my jumping mechanics. Thanks in advance :) Here's the code:

using System.Collections; using System.Collections.Generic; using UnityEngine;

public class FPSController : MonoBehaviour {

 public CharacterController controller;

 public float speed = 12f;
 public float gravity = -20f;
 

 public Transform groundCheck;
 public float groundDistance = 0.6f;
 public LayerMask groundMask;
 public float jumpHieght = 3f;
 bool isGrounded;
 Vector3 velocity;

 // Use this for initialization
 void Start () {
     
 }
 
 // Update is called once per frame
 void Update () {
     speed = 12f;
     isGrounded = Physics.CheckSphere(groundCheck.position, groundDistance, groundMask);

     if(isGrounded && velocity.y < 0)
     {
         velocity.y = -12f;
         speed = 12f;
     }
     

     float x = Input.GetAxis("Horizontal");
     float z = Input.GetAxis("Vertical");

     Vector3 move = (transform.right * x + transform.forward * z).normalized;

     controller.Move(move * speed * Time.deltaTime);

     if(Input.GetButtonDown("Jump") && isGrounded)
     {
         
         velocity.y = Mathf.Sqrt(jumpHieght * -2f * gravity);
     }

     velocity.y += gravity * Time.deltaTime;

     controller.Move(velocity * Time.deltaTime);
 }

}