Unless it's been converted your engine has two ignition modules and a Spark Advance Module (SAM) that tells the ignition modules when to fire. Since you did the ohms test in the manual I'll guess it has not been converted.
There will be seven wires coming out of that SAM. You may have to remove the screws attaching the SAM and pull it out slightly but find the white wire and disconnect it. NOTE. This will disable any safety switches on the equipment so use caution. You will still be able to shut the engine off with the key switch due to the key switch shutting off voltage going to the SAM. If the issue goes away after disconnecting the white wire, the circuit the white wire connects to is grounding out somewhere. If the issue is still present after disconnecting the white wire, check the red wire for battery voltage with the key in both the on and start positions. Also inspect the back side of the SAM for any cracks in the epoxy potting material. If there are cracks in that potting material the SAM is faulty and will eventually fail completely. The SAM is no longer available so you would need to find someone with old stock or possibly find a used one. If you end up needing one, the part number you'd be looking for is, 24 584 09-S. If you can't find a SAM the only other option I am aware of is to get the Kohler ignition conversion kit. That part number is 24 755 307-S. I would opt for the conversion kit if it were me unless you can find a used SAM cheap. The price of a conversion kit will be less than a new SAM and it will get you to a simpler ignition system.
