There are still a few more sources to check, but following up some of the leads in Commodore Worth Moore’s obituaries has already yielded some additional information.
Eldest daughter Dana married Francis M. Stephens in around 1906 and they had at least three children: son Francis Martin (who died young) and daughters Virginia and Mary Janet/Janet Mary. The family lived in Georgia, Tennessee, and Charlotte, North Carolina.
I suspected son T. J.’s initials stood for Thomas Jefferson and found a Thomas J. Moore living in Longview, Gregg Co., Texas (C. W. Moore’s obituary mentioned that T. J. lived in Longview, Texas). He married Lillie/Lilly Carlisle in around 1908 and they had two children, daughter Annie Adele/Adele Annie and son Fletcher. By the 1930 census Thomas and Lilly were divorced. I may have found Thomas J. Moore in 1930 as an inmate in the National Sanitorium in Washington County, Tennessee; I am not sure why he was in Tennessee, but the state of birth, age, and marital status (divorced) fit.