"Meine Familie ist dran, wenn Sie mir nicht helfen!"
I was lucky to find the given translation only by guessing about the additional meanings of "dran" that would make sense here. "dran" is too unspecific here to deduce the meaning "in trouble".
"meine Familie ist dran" would mainly be understood as "It's my family's turn" or "It is up to my family" "or "my family attends to it". Maybe there are even more meanings.
The sense of "in trouble" is mostly used in common expressions like "dann bist du dran" or "jetzt bin ich dran". But even here, it depends very much on accentuation: "Dann bist du DRAN." means "Then you are in trouble." "Dann bist DU dran" means "Then it's your turn".
Wow, thanks so much for this comment. "Dran" was actually one of the first words I learned in German, because I started learning German, so I could play board games in it. :)
Until now, I also didn't know that this can mean trouble too!