On my current playthrough, the closest swamp to spawn is literally 3 days' travel, on foot. Note that the spawn point is the center of the map, and realize that Meadows and Black Forest are simply the most "hubward" (toward the center) biomes. The others are "rimward" (toward the edge).

Get to the ocean. Build a raft karve. Sail directly away from the spawn point to find the other biomes.

Edited to add: You might want to explore East/West, since North and South are supposed to be for the "Deep North" and "Ashlands" biomes, which are not yet in the game.