which LED is this that turned on only once? If you can, use a multimeter to measure the voltage of the battery (unplugged), and THEN plug it in and see how much the voltage drops. If it drops a lot then you probably have a direct short somewhere. Next time this happens (LED goes on only once), it's important to try to collect as much information as possible; feel around for any components that have heated up, (maybe not such a good idea, but...) smell for signs of fried electronics, do a quick visual inspection for neighboring leads that may be touching, etc. Of course the first thing you should do is disconnect power before the damage gets worse.
If it rotated, and then stopped rotating, this could be the signs of an improperly modified servo. I'm not sure how the modification here is being done, but is it possible that the servo has reached the position that the PWM is indicating? (I've hacked a few servos by sawing off the potentiometer knob, leaving potential for the remaining stub to still be turned by the servo...) Someone else answer this.
When you did this last test, touching the ground and signal bus with the multimeter, what setting was your multimeter on? Was your battery connected (ie, was your servo being powered)? I ask this, because without power, a servo shouldn't be able to move, even with the small current a multimeter can provide for continuity testing (it can light an LED, but not much more than that).