Kick,
Cellulostic browning of a wool rug is the result of natural lignin up-wicking from vegetable fibres (most commonly jute) in the foundation of the rug. The two things you need to do to avoid this is to avoid a prolonged drying time and to keep the chemistry on the acid side. A Woolsafe cleaning agent followed by an acid rinse should take care of the latter and if you are to employ an extraction clean, avoiding over-wetting and use of an air-mover should reduce drying times.