I think my next step is to start turning off specific electrical circuits to see if I can determine which circuit may be causing the problem. I'm wondering if a module has gone bad an is causing the issue.
A module going bad should not stop other modules or wall switches from receiving an X10 command. Finding out how many circuits are effected is a great starting point. I predict you will find a noise source (like a CFL bulb). Also are you sure your repeater is still working on both phases? and Will the transceiver click when it is plugged in same outlet where the non working Appliance Module is?