There were also other children:
William (1799), Robert Dugdale (1803), Esther Dugdale (1808), all of whom appear on the opc-dorset site.
William Marshfield, cooper and his son, Robert Dugdale, both appear in the 1830 directory for Dorset, and William's will (he died in 1838 aged 80) is available online for £3.50:
http://www.nationalarchives.gov.uk/documentsonline/details-result.asp?Edoc_Id=355059&queryType=1&resultcount=1A William Marshfield (s Edward and
Elizabeth) baptised in 1758 in Steeple is worth following up as Steeple is only 4/5 miles South of Wareham. It may be that he was baptised a year or two after his birth as there is a will on ancestry for Edward Marshfield of Steeple (1757) that mentions his wife
Elizabeth, father Joseph, brother William, and a few others. I haven't looked at it in detail, but I am sure it is worth doing so. Joseph (1779 aged 77) and his wife 'Jone' appear to have been buried nearby in Coombe Keynes where there a lot of Marshfields.
Richard
Added: there are several online trees with these names in them, one of which at least refers to William as William Francis Marshfield.