Like southern72, I've seen those relays near the batteries give trouble before. There are a few different lots of relays, so I'm not sure which ones you've checked. Does the fuel lift pump work when you turn the key to on?
There is a little black plastic relay (AM53945) attached to the engine block near the starter motor (you need to take out the rear panel from the grain tank to get to it) - have you checked this one? It gave me grief on a 9500.
I've also seen another 9600 with the similar starting symptoms as yours. A circuit breaker in the main electrics box kept getting hot and tripping - long story, but the harness had rubbed through on the chassis, shorting the main power and killing the engine or any attempt to start it.
If any of the circuit breakers feel hot, you may have a similar problem.
The separator not engaging may be part of the above problem, although sometimes the multi pin connector on the right hand side of the cab (accesible from outside just in front of the thresher) can get dry. Cleaning it out with air and some CRC has been known to fix electric control problems.
Just thinking out loud - hope some of it helps ya.