The story about storks bringing babies has nothing to do with swamps, and it was not a matter of belief among the general population. The story is what folklorists call a "fict" - a story told to children to be believed, but which adults did not believe. It is much like Santa Claus, the Tooth Fairy, and the Easter Bunny. I would have put a spoiler alert on this answer as a caution to believers, but I don't know how to do that!

The origin of the story about the stork is in their nests far above ground. I can't imagine the source of the idea that it had to do with swamps, but that bit of speculation is far from what really unfolded.

A common practice related to childbirth resulted in the traditional fict of the stork in western European culture. A woman was in jeopardy during birth because a host of supernatural creatures might attempt to abduct her and leave a replica in her place that would appear to be her corpse. To avoid such a fate, people ushered children outside and sealed the house. When the birth was complete, the doors and windows could be reopened and the children returned. There was a need, however, to explain the arrival of the infant.

Parents wanting to avoid the topic of conception and childbirth relied on the fact that the only opening to the house was the chimney. Coincidentally, European peasants had observed that storks nested on the little-used gallows or “Catherine” wheel in towns. These looked like wagon wheels positioned horizontally on tall poles, and storks found them a safe place to nest. The birds were regarded as good luck, so it was not uncommon for people to construct a similar wheel on top of their roofs to attract storks. When children asked about the origin of the infant, it was easy to suggest that it had arrived by way of the chimney and that the stork had accomplished this deed.