Al,
Actually, the connection to the Ap Adams line was disproven in 1927 by J.G. Bartlett in his Henry Adams of Somersetshire, England & Braintree, Massachusetts. A digital version of this book is available to subscribers of Ancestry.com. You may be able to find a print copy near you through OpenLibrary.org.
On page 21, Bartlett demonstrates how the line of Ap Adams died out in 1424.
Later on, he describes his analysis of parish and probate records, which confirms that the lineage went like this (unless otherwise noted events took place in Barton St. David, Somersetshire):
1. Unknown Adams b abt 1475. Two sons-- Robert (who had no issue) and John:
2. John Adams b abt 1500-1505; m possibly Alys who was widow Alys Adams in 1542/3.
3. Henry Adams b abt 1531 Barton David; died without a will after 1586; son John granted administration of his estate. May have married the Rose Adams who was buried 20 Sept. 1598.
4. John Adams, b abt 1555; buried 22 Mar 1603/4; will dated 19 Mar 1603/4; m Agnes, possibly Stone who was bur 15 Jan 1615/6. He had two sons and some daughters; he was estranged from his oldest son, John, and tried to control how his inheritance would go to the grandchildren, not the son. His second son was Henry Adams the emigrant. [John Schmeeckle: there is no son Jeremiah mentioned who could be brother of Henry.]
5. Henry Adams b abt 1583; named (unsuccessfully) executor of his father's will in 1604. m. 19 Oct 1609 Edith Squire. Records of baptisms of three children in Barton David (1612) and Kingweston (1622 and 1629)
The 1927 genealogy appears to be the best researched and documented history of the immigrant's ancestry, unless something more recent has added further details.
Because Henry did not emigrate until 1638, he is not yet documented by Robert Charles Anderson's Great Migration series, which currently only goes to 1635 or 6.
I therefore recommend following the lineage provided in the 1927 Bartlett book until better information can be located.