Moonglow Bay

To unlock this achievement, you need to finish the story without using the radio on your boat to call for a tow back to town. There are several reasons why you want or need to call for a tow in the first place, which are
- to use as fast-travel back to town.
- when your boat is damaged too much and it won't go any further.
- when your boat is stuck and you can't move it (glitched).
The first one is easy to avoid. Simply don't use the radio to call for a tow when you don't need to, at least until you've reached the end of the story. The other two reasons demand towing to continue the game (and thus void the achievement), so to minimize these risks I have included some tips below.

Damage
During your adventures, you can damage your boat when colliding with walls, islands, large creatures, etc. The boat will show more and more cracks when it receives damage as shown in the image below.

To remedy this, you can buy repairs at the Boat Shop, costing 40 shells for a small repair and 100 shells for a big repair. I recommend buying a repair as soon as possible whenever your boat is damaged, before it is too late and your boat won't run anymore. Further into the game, halfway through chapter 2, you will also be able to buy the Reinforced and Armored Hull from the Boat Shop to reduce the amount of damage to your boat, costing 500 and 300 shells respectively.
You can also use frequent saves to mitigate the risk of damage to your boat. You are most likely to suffer damage to your boat during the first and second 'Boss Fight', so make sure to save before facing them.

Stuck
It is also possible that your boat gets stuck for whatever reason and you cannot move it, so the only way to get out of this situation would be to call for a tow. Especially when traversing in the Astral Clouds, the final biome of the game where you have to fly over walls to reach different areas, this can become a real problem. The only way to fix this is to load up a previous save. Just to be sure, make a save before going into this part of the map so you don't have to redo (a large part of) the game to unlock this achievement.