You've raised one of the harder problems in the Andrew family genealogy.
There's an April 1637 list of people going to New England with "William Andrewes of Ipswich Mr of the John and Dorothy of Ipswich and With William Andrews his Sone Mr of the Rose of Yarmouth."
One of these William Andrews is the shipmaster and merchant who settled in Cambridge, Massachusetts, shortly after this voyage (as he purchased property in the winter of 1637/8).His descendants settled in the New Haven area.
The question is which William is our immigrant.
Choice 1: The immigrant is the elder William.In this case, either (1) the son William died some time after this voyage [without surviving children] and before his father wrote a will in 1651 showing he had only one surviving son named Samuel; or (2) there's a scribal error (17th century typo) and the son is actually Samuel, who would have been about age 18 to 20, which is young but not impossible to be mastering a ship (and we know Samuel was a respected mariner and navigator later in life -- Massachusetts hired him to mark its northern coastal boundary).
Choice 2: The immigrant is the son William, in which case we've learned who his father was.
I've been digging in various records for 30 years and still can't give you an answer as to which is right.What I can tell you is that either (a) William the immigrant was an older man than most people initially guess (the first customs records in Ipswich for William Andrews that I've found are from 1616 -- if that's our immigrant, he's born around 1585 rather than c. 1600 as is often guessed) or (b) the son immigrated and the father, apparently, did not and is the William Andrew who still owns property in Ipswich in 1640 (after the time the immigrant, in his confession to Thomas Shepard, said he'd sold all property in England).
I realize this is frustrating!I hope it is, at least, useful!