Hi Doug,
First off I suggest that you disconnect the positive terminal at the battery and disconnect each electrical connector inspect and clean. The male side of the connectors can be easily cleaned with a small wire brush, the female side not so easily. The following connectors: Engine harness, module connector, brake interlock, LH neutral, RH neutral, mower engage, key switch, start relay, and possibly the fuel pump. Do these one at a time, they are all different and some may need a small screwdriver to release the locks. Do them one at a time, reconnect them and reconnect the battery positive terminal and try to restart and operate the machine normally.
Bear in mind that all the safety circuits feed into the module which compares each and either allows other operations to happen or not. Your module is different from an air cooled machine module as it has input from the engine temp sensor.
Trace all the red wire circuits and you will see that as the power passes thru the safety switches, the wiring changes color as they pass onto and connect to the module plug and into the module. The yellow wire is the charging system output. As you trace it from the engine harness connector, it goes to a fuse holder and on to a terminal that appears to be not connected to anything. This terminal actually connects to a terminal on the keyswitch and when the keyswitch is in the off position, the circuit is open which isolates the charging system. The reason for this is to prevent backfeed into the charging system when the engine is not running which will quickly discharge the battery.
With the battery positive disconnected and the module plug disconnected you can check each safety interlock circuit for proper operation and hopefully diagnose where the problem is. All battery positive power starts at the red/yel wire on the solenoid, thru the fuse, splits off two ways, one to the relay which is the start relay and the other to the ammeter and onto the power into the keyswitch.
I don't know your skill level so this info may be above you.
Have fun!!
Mad Mackie in CT