One of Charles Adams's (son of George C. Adams) sons was named George Wilbur Adams (1867-1943). This George Wilbur Adams was my g-g-grandfather. His wife (my g-g-grandmother) was Winnie May Tindel (1873-1913). They were married in Johnson Co., IN, in the 1890's. After Winnie died, this George then remarried twice more. He died in a town near Pittsburgh.
As you notice from the birth date I have for my George W. and the probable birthdate of your George W., these two really don't match up. So, at this point I don't think they are one in the same.
Although, it is interesting that your George W. Adams's daughter (your grandmother) was born in Noble Co., IN. I believe that my Charles Adams's brother Sylvester S. Adams (1832-1912) lived and died in Noble Co., IN, in the area of Kendallville. So, there might be some connection after all.
Let me know if you need any further information on my Adams family. Possibly I have some piece of information that might sound familiar to you.