A new sensor should be $5-$20 (allowing for Audi prices it is worth $1.50). If a sensor is bad, the light will stay off when the car is started, and then come on after driving the car from 10 feet to 2 miles. If the problem is elsewhere, it may turn the light on before moving the car.
To find the bad sensor, listen carefully after starting your car for unusual noises from one wheel. That's the one with the bad sensor. The noise is the ABS kicking in and out, until it decides to shut down. You can hook up a DMM to the sensor outputs, raise the car and spin the wheel. You get a square wave (A/C) from the sensor. The actual value is irrelevant, matching them is. Basically, you want to get the same reading from all sensors spinning the wheel approximately the same speed by hand.
Try removing the sensor, cleaning it, and reinstalling it until it rubs the hub. DON'T drive it like this -- you may damage it!! You must back it out slightly. It should not rub, but be very close. You can get a little plastic cap from the dealer for ~$0.80 that will set the proper distance for you and keep the sensors form getting eaten.
The sensors too far from the driveshaft will give you pedal "boing" -- this is a groaning/spring sound just as you come up to a stop. The sensors too close will give instant ABS pedal kick in. Too far out tends to shut off ABS quicker than too close. If you can't get the light to go out at all, you might have a fault in the center differential lock switch, or one of the sensors is just bad. Worn or crudded driveshaft splines can do it too. There is an inspection/water drain at the bottom of each strut, you can sometimes get a brass bottle brush in there, though pulling the driveshaft out is easier, and about a 20 minute job.
First check for proper sensor output, then check the differential switch, then the driveshafts. Rarely does that ABS box go bad. Sensors out of alignment usually allow the light to come on till the first brake app or 20-30 seconds, whichever comes first.
