I came across 2 other words for the occupation I know as fuller.
It seems to have been tucker in the south west. Not actually a family member, but related to some other reseach I was doing, William Hayne was master of the Gild of Weavers, Tuckers and Shearmen of Exeter in 1625, having joined the guild in 1606. And he was a tucker.
According to the National Instirute for Genealogical Studies, the word fuller was used in the south and east of England, and wa(u)lker in the west and north.