Try searching the Five Civilized Tribes Application files on Ancestry.com or FamilySearch.com. The files are also on archives.org, but you won't be able to search by name. The Council may provide the information you need, but they may require you to provide proof that you are a descendant.
Dear Ms. Washburn,
Thank you for posting your request on History Hub!
We searched the National Archives Catalog and located a series titled Copy of a Roll of Cherokees in the West, ca. 1909 - ca. 1909 in the Records of the U.S. Court of Claims (Record Group 123) that may contain the information you seek. For access and/or copies of these records, please contact the National Archives at Washington DC - Textual Reference (RDT1) via email at email@example.com.
We also searched Ancestry and located a Dragging Canoe on the 1852 Drennen Roll of Cherokee, the first census taken after the Trail of Tears. Dragging Canoe is listed with his daughter, Nellup. To view these records on Ancestry, there may be a fee. Or you may view the records online at one NARA's facilities for free via ALIC. For the nearest NARA location, please consult our web page at https://www.archives.gov/locations/. Many of the records are also available on FamilySearch for free.
We hope this is helpful. Best of luck with your research!
[Information provided by Cody White, Subject Matter Expert]