My first year on a team I had the same problem as the OP. Our solution was to use the Cherry AN8 sensor. This sensor is a Hall Effect rotational position sensor. It somes in a 360 degree variant, and is analog output. It will output a voltage between 0.5V and 4.5V, so it is easily connected to the analog breakout board. I never had any problems with a dead band on the sensor between the 0 and 360 area.
The reason we love this sensor is that it is absolute between power cycles of the robot, it will always output based upon the magnet's orientation. It will also automatically wrap the voltage at the 0-360 transition, so there is only one voltage per angle (as opposed to the multi-turn potentiometers).
Mechanically it can be mounted similar to the potentiometers, there is a magnetic head which has to be tied in to the rotational system. Then the sensor body gets mounted a few millimeters above that sensor head (it is a non-contact sensing system).
Here is the datasheet for the sensor:
http://www.cherrycorp.com/english/se...AN8_Series.pdf
Here is a link to the Digikey product page for the sensor:
http://www.digikey.com/product-searc...words=CH187-ND
I also had created a C++ class for this sensor, so if anybody is interested let me know and I can dig that out and post it.