Here's what we know about Benjamin Wham - the first known Wham in the U.S.:
 

BENJAMIN WHAM was born ca. 1744 in County Derry , Ireland, and died June 1820 in Chester County, SC. Benjamin was married (Widow Mrs. Woodside- second wife?) in Ireland.   It is not known with certainly that Benjamin's wife accompanied him in the U.S.

Benjamin had two sons: William and Joseph.

Benjamin and his son William arrived at the Port of Charleston in 1797.

Joseph with his family and with his two half brothers, Thomas and James Woodside, came to the United States ca. 1806, having been shipwrecked in Cuba on a voyage from Ireland. They apparenly spent several months in Cuba before going on to South Carolina.

Benjamin purchased real estate in Chester County, SC in 1798.   Details

A family bible at 8 St. Theresa Drive, Charleston, SC, several years ago contained the following information:
"Benjamin Wham settled in Chester, 1797. Joseph Wham Sr. of Ireland settled in Chester 1806."
 

The following information is from the "British Aliens 1812" book:
Wham, Benjamin, age 68, 14 years in U.S.  1 in family, Chester District,
farmer (31 Aug.-8 Sept. 1812).
Wham, Wm., age 35, 14 years in U.S., 8 in family Chester District,
wheelwright (31 Aug.-8 Sept. 1812),.

The dates above do not exactly match other known information, but they are close.
 

Will of Benjamin Wham

"In the Name of God Amen
I Benjamin WHAM of the District of Chester and State of South Carolina
being Sickly and weak in Body but of sound Mind and Memory praised be
God for the same do make and declare this my last will and testament in
Manner and form following

First I will and desire that I may be decently buried at the discretion
of my executors hereafter named as to what worldly property I own
whether real or personal I will that it be equally divided between my
two sons VIZ William and Joseph either to be praised or sold in order
for such a division to be made According as by executors shall deem most
convenient But to be more Explicit I will that my Son Joseph have one
half and my son William the other half and I also nominate Constitute
and appoint my son Joseph and John HUNTER Executors of this my last will
and Testament hereby revoking and making void all and every will hereto-
for made by and do declare this to be my last will and testament

In witness wereof I have hereunto Set My hand and Seal the Eighth Day
of April one Thousand eight hundred and twenty

Benjamin Wham (Seal)

Signed Sealed and Delivered
in presence of us
John TERRY
Stewart BURNS
James McNINCH

Probated June 1820
Recorded Book H, Page 12, Apt. 72, Pkg. 1132"
 

Children of BENJAMIN WHAM and UNKNOWN? are:
 JOSEPH  WHAM, b. 1768, County Derry, Ireland; d. November 20, 1855, Chester County, SC.
 WILLIAM WHAM, b. 1771, Derry County, Ireland; d. July 12, 1833, Lincoln County, TN.

The two sons of Benjamin, William and Joseph, followed different paths.

William and his family migrated to Tennessee and then on to Illinois, forming the " Northern Branch"  of Whams.
Joseph and his family stayed in South Carolina and formed the " Southern Branch"  of Whams.