The transition to a story-driven format in Adventure Time happens gradually over time. Starting from about the middle of the series, the episodes have a stronger narrative focus and a more coherent storyline.
Adventure Time begins to show a more story-driven approach in the later seasons. There's a greater emphasis on overarching story arcs and interconnected events that build upon each other.
It becomes story-driven around season 3. The plotlines start to get more complex and character development takes a central role.