Since your line stems from Charles "IV," my guess is that your family moved through Georgia.We are cousins at a considerable distance since our last common ancestor was Charles "III" -- my family comes down from your Charles "IV"s brother, William.
I know a little bit about the possible West Indies connection.But first I'm a bit surprised by the "1627" date.According to Bill Porter (married to my 2d cousin) there is an entry for an adult Andrew Fallin and the christening of his son (call him Andrew II) on June 3, 1680 in St. Andrews Parish, Barbados. (Hotten, "List of Emigrants to America 1600-1700", 2d. Ed., pp. 470,471)
From a friend in the UK I also have an Andrew "Follin"[likely Andrew II] showing up as aged 37 in St. Andrews Parish in a 1715 census (with children, Andrew, Ann, and Francis) Kent, "Barbados and the Americas", p. 168.
These are good clues to someone who may well have been a relative of the original Charles "I" Fallon/Fallin in Northumberland County VA in the 1640's [his name seems to have been commonly spelled "Fallon" -- with the "Fallin" spelling emerging in the second generation].However, the dates don't line up for ancestor status.
So I am extremely interested in the source for your 1627 date -- if it's positive then we have real progress.
I also have clues to other variant spellings (Fallan, Fallon) elsewhere "in the West Indies earlier than 1680, but let's stay with the Andrews for now.Please let us know where your date comes from.