There is no hum from a IC-706 setup if the cable and strappings are done correct. I guess you have a screen which is broken. Measure the cables to be sure that all screens and wires are connected, and that there is no shortcut between the different pins. Check the red straps with an ohm meter, they may be broken inside the red plastic. If all connections is OK try to use a separate PS for the radio and RRC, or if you do that try to use the same. Ferrits do not help against hum, they may help if you have RFI, but it's always better to keep the RF out of the shack than trying to get rid of it when it's already inside.
73 de mike