No fuel should be flowing thru the main jet into the motor during idle.
as I said check to make sure the timing is advanceing as the extra fuel starts hitting the motor.
or just tak it into a shop to let them fix it
I have never worked on a Suzuki, so no help on wires.
but see if you can find a service manual with wiring diagrams for your motor.
trace the wire from ignition switch to the start relay.
find out where you get 12+ to and where it stops
if you can hear the relay operating when you hit each switch, then you are left with a bad battery cable or just a connection on either end of the + or - cable
most boats have a boat side filter and the one on the outboard, just do not install a small micron filter in the boat side.
get rid of the big stuff and water with the one in boat and let the motors filters handle the rest
