Diane, of all the names you mentioned, there are two William H. Fields, that I found on the Index to Applications Submitted for the Eastern Cherokee Roll of 1909 (Guion Miller Roll), page 100. (NWCTB-75-P1163E589-INDEX). One was in transit and the other was from Kansas. I would go looking for the one with application number: 24634 (I.T.). You should consider when you grandfather William H. Fields was born to determine if applicable. He would had to have been old enough to complete the application sometime before 1909. Now, just because none of your family members are on this list, that only means that they did not want to surface the fact that they were of Cherokee descendants, or were perhaps, unaware.
Recommend you visit these sites. By the way, do not be alarmed at the word Melungeon. My sister and I take it as a positive to determining our roots.http://www.rootsweb.com/~cherokee/miller.html http://dir.infojump.com/dir/dmoz/Society/Ethnicity/Melungeon... http://www.cherokeepub.com/