Well you are coveringf a lot of ground that I am familiar with - Oklahoma and Oregon.
Here is what I suggest you do:
Start with your dad if he is still living. Ask him his parents' names. His date and place of birth, etc.
If he isn't alive, or unavailbale, try your Uncle Jerry or Uncle Mike. Gather all the information you can get - names, dates, places. And then ask them who it is who is supposed to be full-blooded Cherokee - her name, etc.
Since she is your father's mother, her maiden name was probably not Arney - so be sure to find out what it was.
This is the websit that explains how to find out if your family is on the rolls of the Five Civilized Tribes (Cherokee, Choctoaw, Chickasaw, Creek, and Seminole) and enrolled per the Dawes Act.
It takes a lot of time and patience, but you can find the name if it is there!