Cheers -- looks like you took the hard route, which is fine, but the puzzles are definitely 200% harder through the mines compared to the north route. The game warns of this, but I know we gamers are kind of trained by long familiarity to ignore signs that say "danger," heh.
For the Barrel Swamp, the key to this is getting the sledge hammer, then the map. The trick is:
1. Go right from the first screen to the second, and get the sledge.
2. Then make your way north onto the third screen.
3. Smash the two barrels to your left, looking towards where the map is.
4. Walk across the deep water here, which is false, and pick up the map.
5. Holding the map, you can now see all the false water, and can easily make your way to the exit.
Hope that helps, and that your enjoyment increases!