It's a Quattro not an amphibious landing craft!

Audi's are not designed to cross rivers! Driving over puddles more than 6" is not wise as the sudden cooling of the transmission and differential can suck in water through the breather tube! I am surprised that you didn't stall in the puddle itself as I find the ignition electronics too vulnerable.
Where you driving in NJ on Rt3 last night, a notorious instant reservoir that the state refuses to fix. Actually we got quite a few of these miracle highways Rt 3, Rt 17, and Rt 22 just to name a few.
Does the car roll in neutral? With the engine off can you shift through the gears? If either the pilot bearing or the throw out bearing is worn and low on grease then even a minute amount of water will cause it to rust. I think the problem might be with the pilot bearing. I know first hand that a bad pilot bearing has a tendency to make shifting gears difficult if not impossible. Either one is going to be a bitch