Verb for "to become bog/marshland/swamp"
Solution 1:
How about inundate? From M-W:
inundate: to cover with a flood, overflow
Synonyms include drown, flood, submerge, submerse, swamp. Inundate is more general than flood.
Marshlands, bogs, and swamps are all the result of an area being inundated by water for one reason or another, e.g., tides, seasonal flooding of low-level areas, precipitation, etc.
To inundate is the opposite of your to drain.