Bartlett's book "Henry Adams of Somersetshire, England and Braintree, Mass." published in 1927 put to rest firmly (but sadly not finally) the connection between the ap Adams pedigree and the Henry Adams family. On page 21 he gives the complete legitimate pedigree of Ap Adams and shows that it "daughters out" in about 1380.This is after only 5 verified generations from the founder.So those Georges in later generations are bogus.
The first mention of the name George in the family of Henry Adams and Elizabeth Squire is their grandson George born to Thomas Adams and Mary ? in Concord, MA on 29 May 1650.
You mention that your George was "of Watertown" and born about 1620.When was he in Watertown?I may have some resources that could help but I need more dates.