Problem with free 3D - movement script

Hi. This is kinda hard to explain but I´ll try.

I´m creating a sort-of space shooter which requires free movement in all 3 dimensions. I want the movement to be like in Dark Souls, but in 3D space.

I got everything running for moving in the XZ-Plane but I can´t figure out how to do it correctly for 3 dimensions.

As soon as the camera looks up or down and I move forwards or backwards, everything seems alright but as soon as I try to move left or right, everything becomes a mess.

Here´s my controls script:

 class Controls extends MonoBehaviour{
 
     public var moveDirection:Vector3;
     public var moveSpeed:int = 10;
     public var cameraTransform:Transform;
     public var inputParser:InputParser;
     
     public function Update():void{
         this.freeMove();
         this.genericMove();
         this.cameraTransform.position = this.transform.position;
     }
     
     public function freeMove():void{
         var right = this.cameraTransform.right;
         var forward = this.cameraTransform.forward;
     
         var v = this.inputParser.verticalLeftStick;
         var h = this.inputParser.horizontalLeftStick;
         moveDirection = v * forward + h * right;
         moveDirection = moveDirection.normalized;
         if(moveDirection != Vector3.zero){
             transform.rotation = Quaternion.LookRotation(moveDirection);
             transform.Translate(Vector3.forward * Time.deltaTime * moveSpeed);
         }
     }
     
     public function genericMove(){
         this.GetComponent.<Rigidbody>().velocity = Vector3.zero;
         if(this.inputParser.yHold){
             this.transform.localPosition.y += this.moveSpeed * Time.deltaTime;
         }
         if(this.inputParser.aHold){
             this.transform.localPosition.y -= this.moveSpeed * Time.deltaTime;
         }
     }
 }

And here´s the Camera controls script:

 class CameraControls extends MonoBehaviour {
     public var rotationSpeed:int = 10;
     public var target:Transform;
     public var normalAnchor:Transform;
     public var leftAnchor:Transform;
     public var rightAnchor:Transform;
     public var cameraAnchor:Transform;
     public var inputParser:InputParser;
     
     public function LateUpdate(){
         this.parseInput();
     }
     
     private function parseInput(){
         var rotationY = this.inputParser.horizontalRightStick * this.rotationSpeed * Time.deltaTime;
         var rotationX = this.inputParser.verticalRightStick * this.rotationSpeed * Time.deltaTime;
         this.cameraAnchor.transform.Rotate(rotationX, rotationY, 0);
         Camera.main.transform.position = this.normalAnchor.position;
         Camera.main.transform.rotation = this.normalAnchor.rotation;
     }
 }

What is the trick to make it work right? I can provide an example project with this if needed.

EDIT #1: I also noticed that the camera rotation isn´t working correctly either.