Welcome to RootsChat Andrew.
I've found only one James BOND baptized in Suffolk c.1776: a son of John and Elizabeth in 1777 at Redgrave with Botesdale (close to the Norfolk border, nowhere near Woodbridge or Ipswich).
Ancestry has a register of indenture duties (IR 1/33) showing that a James BOND was apprenticed in 1788 to a Liverpool merchant, John SPARLING, whose goods may have been shipped across the Atlantic, but there is no sign of either seamanship or Suffolk origins for James.
I see no relevant bastardy order at
Suffolk Archives but I suspect that Ann DUNNETT (buried at Woodbridge in 1794 aged 38) was a daughter of Edmond or Edmund BOND and Anne GARNHAM (m. Walton 1751) whose children were christened in various parishes:
1753 Edmund at Walton [buried at Hemley (a young man) in 1775?]
1754 Robert at Kirton
1756 Anne at Bucklesham
1759 Joseph at Bucklesham
1762 Sarah at Bucklesham [buried at Hemley 1764]
1768 Sarah at Hemley
It may be significant that the names Joseph and Robert were given to sons of Daniel and Anne DUNNETT at Woodbridge in the 1780s.
Also buried at Hemley were the clerk [former rector?] of that parish, Edmund BOND, aged 72 in 1778, and Hannah, wife of Robert BOND, aged 29 in 1779.
Robert seems to have married three times:
1779 Hannah CLARK at Hemley
1780 Mary NICHOLLS at Hemley
1794 Elizabeth COCKRELL (a widow) at Nacton
Joseph BOND married Frances BURROWS (a widow) at Woodbridge in 1792. Burials there indicate that he had been the innholder of "The Anchor" before her death in 1814 aged 76 and that he died at the age of 73 in 1832.
Daniel DUNNETT (c.1738-1823) married Elizabeth BROWN at Alderton in 1761 and Frances ROSE (a widow) at Melton in 1797. Since his will (which I can't find) makes no mention of James BOND, I hope you will have better luck with probate records of Ann's other relatives.
Alternatively, you may want to trace the family of Susan DUNNETT who married Thomas BOND at Wherstead in February 1767. A son, Thomas BOND, was baptized there three months later and a daughter, Sarah BOND, at Tattingstone in 1772. Susan died a pauper aged 59 in 1787 and was identified in Tattingstone's burial register as the wife of Thomas BOND and former wife of Joseph DUNNET, née KING. Boyd's marriage index mistakenly names her Sarah if she was the bride of Joseph DUNNET at Tattingstone in 1747. He was apparently buried there in 1763. Their daughter Ann DUNNET (bapt. Tattingstone 1754) may have been called Ann BOND after 1767 but I don't suppose her DNA would carry evidence of a patrilineal link, even if she was in Woodbridge marrying Daniel DUNNETT in 1779. She looks more likely to have married Henry GOSHAWK at Wherstead in 1778.
David