Oh, I understand where you are headed now. I know this much. The Lawing family was a large one and it is possible there could be another William Ambrose or Ambrose Lawing. I have seen where some people didn't use their full names always. William Ambrose Lawing MAY have been full-blood, but the reason he doesn't show up on the Guion-Miller Roll or the 1924 Baker Roll is he died before those rolls came into being. The few Lawings I see listed (I have both books)are in the section of "Applicants Not Eligible". No reason is given.
If you are seeking proof of Cherokee heritage, you may never find out. I don't mean this unkindly. Many did not enroll and those that did often had to prove it, especially if they were less than full-blood. Some applications were appealed and then accepted and others were rejected. And for whatever reason or reasons those that decided these things could/would reject claimants. Today the requirements for becoming a tribal member of the Eastern Band of Cherokee are very strict. Good hunting.
My great-great grandfather was William Benjamin Fleming. There were there 3 men with the exact same name and I had to track all of them down before I could say "this one belongs to me." He rode with the 13th GA Calvary. You might check and see if W.A. Lawing's widow got a pension.