That looks like Sclerotinia sclerotiorum . It is a fungus That does hang around when the soil is infected. http://pnwhandbooks.org/plantdisease/cabbage-and-cauliflower-brassica-sp-sclerotinia-stem-rot-and-watery-soft-rot . Other possibilties include Phytophthora root rot (Phytophthora megasperma). This one is more pronounced on roots but affects the stem at ground level. Black rot norammly affects leaves and heads but can spread to stems. Black leg (Phoma lingam) is like a dry stem rot. http://www.clemson.edu/extension/hgic/pests/plant_pests/veg_fruit/hgic2202.html

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The roots look fine to me, though I am not an expert on brassica roots.... Older leaves are yellow and spotty but this is what always happens....

The Sclerotinia sclerotiorum looks like it makes white fluffy mold... I am not seeing that.

I guess we can rule out over watering as the cause of root rot in your case.