Looks like your engine has smartspark, I looked up your engine and it said that you did not, so I didn't suggest it.
That is very likely your problem
Kohler is not very forthcoming with what system was used on this engine. Most parts diagrams don't show the SAM unit, but the DSAM is shown on the Dealer parts site, but not the ASAM It shows the ASAM compatible ignition modules with a serial break of below 36339000001, but not the Analog speed advance module It also shows the Speed advance module for after serial number 3633900001 but not the 3 prong ignition modules themselves. They supercede directly to the MDI modules.
It looks like the OP has the ASAM Which uses the two terminal ignition modules. The ignition modules for the ASAM system are available but cannot get the ASAM advance module itself. Problems with the advance module are required to convert it to MDI for this engine which requires the replacement of the flywheel. Conversion kit 24 755 308-s which includes both new modules, and a new flywheel.