Drowning is out of question, for a few reasons.
First, the word translated as water-barrel in Russian original is кадушка; a vessel holding perhaps four or five gallons at most. It physically impossible to drown in it.
Second, the entire story is in past tense, except that Groholsky
is staying with Bugrov to this day
is in present. This surely means that he is alive by the time the story is told.
The exact symbolism of the final sentence is not clear; we can only guess why Chekhov mentioned it. It could be that Groholsky is about to burst in tears and needs water to wash his eyes (very melodramatic, but in line with his character). It could be that he has a hangover (Russian tradition associates the sudden attack of thirst with hangover). It could have no hidden meaning at all.