There is a plug on the Aladdin "Video Coach Systems Interface" for the turn signals (that controls the camera switching on the monitor). I would start there, as it's pretty easy to get to. Check the plug connection. You could connect a voltmeter and test that the turn signal voltage is correct when the signals are on. If your getting low voltage or intermittent voltage as the turn signals are operated, that would tell the problem is in the wiring and not in the Aladdin unit.
However, since you say that rain causes the problem 100% of the time, I would think that you have a poor connection somewhere outside of the coach (at the turn signal light itself or one of the wires from the turn signal lamp is exposed).