There are several groups of GAW's in America.
My ancestor John Gaw, Sr. was probably Scot or Irish and was in South Carolina when his son Thomas GAW was born in 1777 (Thomas m Nancy HOOD in 1799 and moved to TN). John Sr. fought in the RW, died by 1788 and his widow Violet m John Calhoun and moved to Indiana with her son John Gaw Jr b 1786. John GAW and Violet also had a daughter Mary b 1784 who m a STROUD and also moved to Indiana.
There were 3 GAW brothers who came over from Enniskillen, Ireland and settled in the Sheneadoah area of Virginia. They were there for a couple of generations but are not my family.
There was a GAW family in Maryland and Pennsylvania in the 1700's. There was a RW soldier among them, his name started with C. I don't know if they are related to my family or not.
There is also a group who came to America through Canada and they may be the family that settled in the Michigan area.
I would be willing to share info with anyone who in interested in the GAW surname. There was also a McGAW family in South Carolina the same time as my ancestor, a William and a John. If they are the same family as mine, the connection is much earlier than the mid-1700's; John GAW b 1786 and John McGaw b about 1784 have been "merged" by some researchers but are not the same person.
Good hunting and please let me know if I can be of help to you.