Two widgets swapping places on drag and drop in Flutter

I am making a puzzle with hexagonal tiles that the user can rotate and reorder to reach the solution. I would like the tile that is dragged and dropped and the tile that is originally in the dropping position to swap places. I guess that I should use the Draggable class wrapped in DragTarget, but I didn't manage even to have the DragTarget accept the Draggable, let alone that there is a widget swap. Here is the code (now with only two tiles in one row, but there will be more in three rows):

import 'package:flutter/material.dart';
import 'package:hexagon/hexagon.dart';

class Puzzle extends StatefulWidget {
  const Puzzle({super.key});

  @override
  State<Puzzle> createState() => _PuzzleState();
}

class _PuzzleState extends State<Puzzle> {
  bool isOnTarget = false;
  String imageName = "";


  @override
  Widget build(BuildContext context) {
    return Scaffold(
      body: Center(
        child: Column(
          children: [
            Row(
              children: [
                DragTarget<String>(
                  builder: (context, candidateData, rejectedData) {
                    return PuzzleTile(image: isOnTarget ? imageName : "assets/images/1.png");
                  },
                  onAcceptWithDetails: (DragTargetDetails<String> image) {
                    setState(() {
                      isOnTarget = true;
                      imageName = image.data;
                    });
                  }, 
                ),
                DragTarget<String>(
                  builder: (context, candidateData, rejectedData) {
                    return PuzzleTile(image: isOnTarget ? imageName : "assets/images/2.png");
                  },
                  onAcceptWithDetails: (image) {
                    setState(() {
                      isOnTarget = true;
                      imageName = image.data;
                    });
                  }, 
                ),
              ],
            )
          ],
        )
      )
    );
  }
}

class PuzzleTile extends StatefulWidget {
  final String image;
  const PuzzleTile({super.key, required this.image});

  @override
  State<PuzzleTile> createState() => _PuzzleTileState();
}

class _PuzzleTileState extends State<PuzzleTile> {
  double angle = 0.0;
  late final String image;
  void _changeRotation() {
    setState(() => angle += 1.0 / 6.0);
  }
  @override
    void initState(){
      super.initState();
      image = widget.image;
    }

  @override
  Widget build(BuildContext context) {
    return Draggable(
      data: image,
      feedback: HexagonWidget.pointy(
        color: Colors.black,
        width: 100,
        elevation: 8,
        child: GestureDetector(
          child: AnimatedRotation(
            turns: angle, 
            duration: const Duration(milliseconds: 200),                   
            child: Image.asset(image)),
          onTap: () {
            setState(() {
              _changeRotation();
            }); 
          },
        ),
      ),
      child: HexagonWidget.pointy(
        color: Colors.black,
        width: 100,
        elevation: 8,
        child: GestureDetector(
          child: AnimatedRotation(
            turns: angle, 
            duration: const Duration(milliseconds: 200),                   
            child: Image.asset(image)),
          onTap: () {
            setState(() {
              _changeRotation();
            }); 
          },
        ),
      ),
    );
  }
}