Since it has a power cable plug on the unit and not a cradle type connection, 1st Doublecheck your connections at the battery and any inline fuse holder for a possible faulty connection.
If problem still persist after ruling out connections, you suspected a possible problem with the transducer cable, try unplugging the transducer cable from the unit and power it up. If it still continues to go off, I would suggest ordering a new power cable and see if the problem goes away. Shouldn’t cost that much for a cable. If it doesn’t fix your problem then more than likely it’s in the unit. you have narrowed it down to the unit by process of illumination. If its only 4 or 5 months old it should be under warranty. Garmin will repair it or just replace it.