Transmission engages but the car will not Move?

I have the car listed above that is having transmission problems. Here's some background:
3.0 V6 engine. Runs great, and up until recently, shifted fine. After driving home one day, we get in to leave and the car will not move in any gear - as though it was in neutral. No noise - no grinding, whining, etc. No leaks either. When I shift into reverse or drive, I can feel the engine shudder slightly (like normal) as it engages, but the car will not move. The transmission fluid level is correct and slightly brown, but not burnt. No metal pieces are in it either - just regular light wear dust. I checked the shift lever (Transmission Range Sensor) on the transaxle and moved it by hand while my wife sat in the driver's seat - same result. The engine will move as reverse or drive gear(s) engage, but the car won't move even if revved up pretty high. As an experiment, I unplugged a connector on the top of the transaxle, near the TRS (see picture of crudely circled plug) (I think this is a 'bulkhead' connector?) And then the car would drive forward but only sometimes in reverse. In forward, it is very, very sluggish, as though it is starting in high gear?