As for the exit stamp - there are always manned booths at Changi, but pax are directed towards the machine gates. You could just say that you need to go to a manned booth. And from my experience there they stamp you out without asking.
JB - if the stars line up, you could get much faster from Changi to JB flying via KL than overland

There are also a lot of chauffeured services from Changi to JB, where you pass immigration/customs without getting out of the car.