Incorrect usage of Vector3.Dot?

Hello, I am trying to make a game similar to Rimworld or Prison Architect, so I have a AI controlled sprite character with 4 different sprites for each main direction and I'm currently trying to calculate the target direction via Vector3.Dot and then assign the correct sprite:

 void Update()
 {
         float dotUp = Vector3.Dot(_navMeshAgent.destination - _thisTransform.position, Vector3.forward);
         float dotRight = Vector3.Dot(_navMeshAgent.destination - _thisTransform.position, Vector3.right);

         if(dotUp > 0.8f)
         {
             // walking up
             spriteObject.GetComponent<SpriteRenderer>().sprite = sprite_Back;
         }
         else if(dotUp < -0.8f)
         {
             // walking down
             spriteObject.GetComponent<SpriteRenderer>().sprite = sprite_Front;
         }
         else if(dotRight > 0.1f)
         {
             // walking right
             spriteObject.GetComponent<SpriteRenderer>().sprite = sprite_Right;
         }
         else if(dotRight < -0.1f)
         {
             // walking left
             spriteObject.GetComponent<SpriteRenderer>().sprite = sprite_Left;
         }
  }

The problem is that if the character has to change direction from left to right or vice versa and if he runs a bit up or down diagonally, he ends up looking up or down first before watching right/left. Is there a way that I can use the Vector3.Dot product to make it behave as it should?

Here's a Web Player demo which should clarify what I mean: WebPlayerDemo