Is there an app on Cydia that will let me turn off dock audio?
Unfortunately, this would require some very specific code that goes beyond anything I've seen on Cydia. That's not to say that a solution does not (or will not by the time I finish this post) exist. However, through my research and experience, this is handled by code much deeper in iOS than most Cydia apps care to touch. I wish I could offer a solution, but unfortunately, I think the only "real" solution is repair or replacement. Sorry. :(

Good luck and happy listening!