The underlying concept is: The waterway was sombre. "To flow sombrely" would describe the manner of flowing, but there is no sombre way of flowing or a happy way of flowing. Water can flow slowly, sluggishly, fast, but not sombrely.
The problem of adjectives after verbs in adverb position is very tricky and it takes some time to get a feeling for this problem. The school rule "after a verb follows an adverb" is really only a school rule. There should be added in brackets: (But there are a lot of cases where an adjective follows because the intent is not to describe the manner.)
Edit: I looked up the etymology of sombre at etymonline. The meaning is 'gloomy, shadowy' from French and ultimately from Latin sub (= under) and umbra (= shadow). So we could clarify the use of the adjective sombre after flowed in this way:
The water flowed (along) as under shadow. An expression made plausible as there are things that dim the light: an overcast sky and a dark cloud at the horizon and that fits with the last words: leading into the heart of darkness.