I've done some additional searching on this family. It appears that the Cherokee claims come from an application filed by descendant Thomas P. Sawyer for a share of the 1907 Eastern Cherokee payment. Here's a link to the application: https://www.fold3.com/image/221210737
The application was rejected as there was no evidence that any of his ancestors had any connection to the Cherokee. Unfortunately, this application was one of tens of thousands filed by white people who had no Cherokee connection. The U.S. government approved a payment of 4 million dollars to the Cherokee in 1907. 90,000 people made claims to a share of the money, but only 30,000 turned out to actually be Cherokee or descendants of Cherokee. Guion Miller, the man in charge, did an extraordinary amount of research to determine which claims were valid. The applications have a wealth of genealogical information, but the claims are false. Attorneys advertised for clients, had a stable of paid people who made affidavits, and made tons of money off the applicants who of course got nothing.