Pathfinding in a Grid-Based RPG... Theorycrafting and solution.

I am working on a Grid-Based title similar to Legend of Grimrock and/or Eye of the Beholder. In terms of pathfinding, I have gone through many issues as of lately.

My enemy has a wander AI which is very incomplete and only for testing purposes; I will implement a second-based AI later (this will be clear when you see the script below).

I tried using A*, but it merely causes the enemy to drag along, and not move in a grid. I had a script that rounded the current position to a whole number, which is the basic grid my game runs on. That worked to an extent; however, the enemy just dragged on without rotating at corners or going through his normal movement behavior.

Is there any way I can have the enemy move in a grid-based functionality, tracking down the player dynamically, and rotating at corners/rotating to face him?

Thanks in advance! :D

Enemy movement script (with wander AI(As you can see, the FoundPlayer method is empty as I have no clue what to do :P)):

 using UnityEngine;
 using System.Collections;
 
 public class EnemyMovement : MonoBehaviour {
     
     public static bool CanMoveForward;
 
     public bool FoundPlayer;
 
     //Translation:
     float movSpeed = 4.0f;
     Vector3 pos;
     Transform tr;
     public static bool moving = false;
     
     //Rotation:
     public static bool rotating = false;
     public float rotSpeed = 360f;
     float rotDegrees = 0f;
     Quaternion rotToAngle ;
     
     void Start () {  
         pos = transform.position;
         tr = transform;
 
         CanMoveForward = true;
 
         FoundPlayer = false;
     }
 
     void MoveTowardsPlayer() {
         return;
     }
 
     // Update is called once per frame
     void Update () {
 
         int randomValue = Random.Range (1, 100);
 
         Debug.DrawRay(transform.position, transform.forward, Color.red);
 
         if (!moving && !rotating && !FoundPlayer) {
 
             if (randomValue == (1 | 2) && tr.position == pos && CanMoveForward) {
                 pos += transform.forward;
                 moving=true;
             }
 
             else if (randomValue == 5 && tr.position == pos) {
                 rotDegrees -= 90f;
                 rotToAngle = Quaternion.Euler(0, rotDegrees, 0);
                 rotating = true;
             }
 
             else if (randomValue == 6 && tr.position == pos) {
                 rotDegrees += 90f;
                 rotToAngle = Quaternion.Euler(0, rotDegrees, 0);
                 rotating = true;
             }
 
         } else if (FoundPlayer) {
             MoveTowardsPlayer();
         }
         
         
         
         //Translation:
         if (moving) {
             if (Vector3.Distance(transform.position,pos) <0.05f){
                 transform.position = pos;
                 moving=false;
                 //                print ("FINISHED MOVE");
             } else {
                 transform.position = Vector3.MoveTowards(transform.position, pos, Time.deltaTime * movSpeed);
             }
         }
         
         //Rotation:
         if (rotating) {
             if (Quaternion.Angle(transform.rotation,rotToAngle) <10f) {
                 transform.rotation = rotToAngle;
                 rotating=false;
                 //                print ("FINISHED TURN");
             } else {
                 transform.rotation = Quaternion.RotateTowards(transform.rotation, rotToAngle, rotSpeed * Time.deltaTime);
             }
         }
     }
     
 }
 

:D