Somber is correct. It’s a complement. Whether to call it a subject complement, predicative adjective, or something else is a matter of debate. It’s also an unusually clever and poetic use of English grammar.
Here is a similar, more ordinary example of this construction:
Rinse the shirt under cold running water until the water runs clear. [Source]
The word somber
The primary meaning of somber is: somewhat dark, as if under a shadow. This agrees with overcast sky. But a more common meaning of somber is a certain melancholy, solemn mood. The water doesn’t have a mood, of course, but I’m sure Conrad chose this word carefully, since the characters were surely in a somber mood after the death just discussed and the lie just told to the deceased’s fiancée. The mood isn’t part of the literal meaning of the sentence, but that meaning will affect a reader.
The grammatical construction
It’s ambiguous whether the water became somber as it flowed—that is, led into darkness—or whether it was somber everywhere. Copperkettle’s answer provides linguistic terminology for these two kinds of complement, and mentions that scientists are still figuring out precise categories for them. However, somber in this sentence calls upon both grammatical meanings simultaneously. One meaning is echoed by under an overcast sky and the other meaning is echoed by lead into the heart of an immense darkness. To pick either “the state of the water” or “the shade that the water took on as it flowed” would miss the deliberate double use of the same complement, which gives the sentence its unique and memorable structure.