I have a John Adams (11/25/1837, died 02/19/1900) buried in Seward Cemetery, Logan Co OK. He is listed as "Brother of Margaret Beam, 45th PA Vol Inf."
No direct connection to William, but interesting nonetheless.
I have a Sarah E. Adams, my husband's great-grandmother, who was born somewhere in PA (still unknown where). Her parents were listed on the 1820 as both born in Ireland, BUT her father's nationality is listed as Scottish, mother's as Irish. That would make her father, Someone Adams, as being of Scottish descent.
If my information sparks anything for you, get in touch, either here or direct through e-mail.
Hope you're still logging on once in awhile to receive this message.