~てしまう and ~でしまう become ~ちゃう and ~じゃう respectively in colloquial speech, ~ちまう and ~じまう are also possible if you want to sound like a middle-aged man. ~てしまう usually means that you regret the action, show embarrassment or disappointment or that it happened by accident, but it also means, as you said, that something is completely done, meaning, for example, 本を読んでしまう to finish reading a book. In your sentence it's the meaning you quoted: to finish. ''Finish drinking mom's cocoa!'' or ''Drink up mom's cocoa!''.