10A!!! IMO A 1N400X (X is voltage spec, 1N4001=100PIV etc) 1A diode is more than enough and easier to deal with than a 10A type, solder wire to the leads and cover it with shrink tube.
You should not need diodes though since you have separate idiot lights. Diodes are needed with all-LED indicators for a single idiot light.
I don't know where the left and right circuits can connect... the obvious place is the single idiot light but you have removed that, presumably.
Maybe the switchpod is somehow leaking current between the signal outputs? If you have markers, maybe there is a wiring error with them that somehow does it? Using a normal bulb connected to one side and flashing the other, you will get an idea how good the cross connection is - flashing bright means there is a good solid cross connection, no or almost no glow from the filament says the cross is leakage: maybe in the switch or where the single idiot light connected.