Yes, Joe ... I, too, have researched this particular BEAIRD family. I'm a FULTON hunter and I suspect your research will help me prove the relationship between two Fultons at Kaskaskia ... James Fulton (d bef 1818) and John Fulton (d bet 1818-1820).
I can prove that John Fulton was born 1747 and acquired land in the Holston Settlement (present TN) as early as 1773 and remained in this area through mid 1801. John Fulton was present at Kaskaskia by Nov 1801.
James Fulton is more difficult. He could be contemporary to John, or could be an older son. Your research may resolve the matter.
In 1809, James Fulton married Eleanor Beaird. Eleanor died before 1811 and James Fulton died before 1818. Could Eleanor be the widow of John Beaird????