How can I make an object stop all momentum and hold it's position in air?

I'm creating a 2D platform game and I have a character that I want to be able to move and jump as you'd expect in a platformer game, but then on the right surfaces be able to switch to climbing. When climbing they would not be affected by gravity and could move in all directions, not just left/right. I was able to make that work as expected, but I noticed that if the object is falling when I switch to climbing they still fall as if they were being affected by a weaker gravity. What I'd like is for the character to completely stop when the button is hit and no longer have gravity affect them at all.
Below is the script I have currently for this character that can attach to walls.

 using System.Collections;
 using System.Collections.Generic;
 using UnityEngine;
 
 public class LizardScript : PlayerCharacter
 {
     public CatScript cat;
     private float climbingSpeed = 7.0f;
     private bool nearWall = false;
     private bool onWall = false;
 
     // Start is called before the first frame update
     void Start()
     {
 
     }
 
     // Update is called once per frame
     void Update()
     {
         if (GameScript.mainPlayer == "lizard")
         {
             startClimb();
             if (onWall)
             {
                 GetComponent<Rigidbody2D>().gravityScale = 0;
                 climb();
             } else
             {
                 GetComponent<Rigidbody2D>().gravityScale = 2;
                 baseCharacter();
             }
         }
     }
 
     void climb()
     {
         var climbing = new Vector3(Input.GetAxis("Horizontal"), Input.GetAxis("Vertical"), 0);
         transform.position += climbing * climbingSpeed * Time.deltaTime;
     }

     void startClimb()
     {
         if (Input.GetKeyDown(KeyCode.G) && nearWall == true)
         {
             onWall = !onWall;
         }
     }
 
     private void OnTriggerEnter2D(Collider2D collision)
     {
         if (collision.gameObject.tag == "Wall")
         {
             nearWall = true;
         }
     }
 
     private void OnTriggerExit2D(Collider2D collision)
     {
         if (collision.gameObject.tag == "Wall")
         {
             nearWall = false;
             onWall = false;
         }
     }
 }
  

Below is the parent class for the LizardScript above. This script is where the jumping and standard movement is handled:

 public class PlayerCharacter : MonoBehaviour
 {
     public float speed = 5.0f;
     public Vector2 jumpHeight;
     public bool onGround = true;
 
     // Start is called before the first frame update
     void Start()
     {
         this.tag = "player";
     }
 
     // Update is called once per frame
     void Update()
     {
 
     }
 
     public void baseCharacter()
     {
         run();
         jump();
     }
 
     public void run()
     {
         var move = new Vector3(Input.GetAxis("Horizontal"), 0, 0);
         transform.position += move * speed * Time.deltaTime;
     }
 
     public void jump()
     {
         if (Input.GetButtonDown("Jump") && onGround == true)
         {
             GetComponent<Rigidbody2D>().AddForce(jumpHeight, ForceMode2D.Impulse);
         }
     }
 
     private void OnCollisionEnter2D(Collision2D collision)
     {
         if (collision.gameObject.tag == "Ground")
         {
             onGround = true;
         }
     }
 
     private void OnCollisionExit2D(Collision2D collision)
     {
         if (collision.gameObject.tag == "Ground")
         {
             onGround = false;
         }
     }
 }

I thought the GetComponent<Rigidbody2D>().gravityScale = 0; in LizardScript would've been enough, but clearly it isn't. If there's anything that I might be missing let me know! Thanks in advance for your time and help!