You can hand wash the Absorber with Dawn with good results. You do need to spend some time rinsing it, though.
FWIW, if your absorber is getting dirty when you dry your car, you probably didn't do good enough job washing it. There shouldn't be anything left for the absorber or a towel to dry but clean rinse water.
IMO, drying a vehicle that isn't totally clean is one of the biggest causes of marring of paint.
Charles