Did you find Edward's death record? Edward died in Edgartown in 1914, age 79. His death record gives father as William P. Chadwick b. Tuckernuck Island, MA [Nantucket]; mother as Sarah Mayhew, b. Edgartown.
In the Edgartown VRs is a marriage for William P. Chadwick and Sally P. Mayhew, May 31, 1832.
Again in Edgartown, a death on 18 Feb 1883 for a William P. Chadwick, age 72, son of Anthony & Mary Chadwick both of Edgartown.
Taking a quick leap here (i.e. needs to be verified), an Anthony Chadwick's death in Nantucket in 1835 says "Anthony, 'from the Vineyard,' h. Polly (d. William Perkins and Eunice), 12th, 9 mo. 1835, P.R.38. [generally debilitated, Dec. 9, a. 53 y. 1 m., at the asylum, P.R.62.]"
And just to a possible twist to this, there is a death in the Edgartown VRs which reads:
"Chadwick, Anthony, "a Portugee a State Pauper," universal decay, [Mar.] 26, 1820, a. 67. PR."
That last tidbit, I have no idea if he's related or not. But all of it gives you a few more places & dates to check. FWIW, Edward and William (as well as an Edward D. Chadwick found randomly in the Marthas Vineyard VRs) were blacksmiths, so not seafaring men, which makes them easier to track.