More research yesterday turned up another possibility for Lulu/Lula Burgess’ parents. After searching the 1880 census for Joseph Haley’s information I found a Lula Hadley, daughter of Joseph and Emma Hadley living in Verona, Oneida, New York. The age of Lulu/Lula was close enough from the census forms. I also knew that Joseph had been married three times according to the 1910 census.
This would mean that Gretchen, Bertha/Beatrix and their other sister were step sisters to Lulu/Lula, not full sisters as I had originally thought. Their father was William Richard Saggesser, who was born about 1847 and died February 19, 1892. Their mother was Clarissa Elizabeth Downs.
So far the one missing piece for Joseph Hadley is the 1900 census, which would have been the census before having married Clarissa in 1906.
It also leaves a big question as to who was Charles Carmon and why had she referred to him as her father.