My front loading GE washer WSSH300 will not drain.

Don't know if this will help or not, but your owners manual is available at http://genet.geappliances.com/DocSearch/Dispatcher
Water won’t drain Drain hose is kinked or • Straighten drain hose and make sure washer is not improperly connected sitting on it. • Top of drain outlet should be less than 8 ft. (2.5 m)
above floor. Household drain may • Check household plumbing. You may need to call be clogged a plumber. Plumbing this washer directly to a pipe is hard on the motor. Usually the drain hose elbow is inserted in a stand pipe, meaning the washer would drain water on the floor is the stand pipe is clogged.
Several items can cause the lack of draining.
If you can get to the drain hose, try lowering to a point that it will drain into a bucket. If it drains the system is not clogged. If you advance the timer and the washer drains this maybe the culprit. The motor can overheat and shut off. I would advance the timer to spin and listen for a hum in the motor and then quiet, a sign that the motor has overheated. If the water drains it sounds like the timer is at fault.
I would first make sure the drain hose is properly installed and not kinked somewhere. In earlier GE's the water drain pump could clog, your manual doesn't indicate that as a defect. Regardless timer, motor and pump are in the cycle to empty the washer water.