Sorry, you asked a question and got a lot of answers, but not what you were looking for. I'll answer as much as I can, but will likely leave you looking for new answers,
John Addison Cady b. 1845 d. 1877
Married Louise Sherman b 1847 d 1892 M abt 1870 Daughter of David L Sherman and Olivia Marie Griggs (Remarried 1878)
Child Harrison "Harry" Howard Cady b. 25 May 1873 in Charlestown.MA d 1927 Putnam CT
Married Margaret Ellis Jackson b 1877 d 1953
John was the son of Elijah Cady & Martha Adams Turner listed in Descendants of Nicholas Cady of Watertown MA by Orrin Peer Allen. Can be found on Google or in re-print with Higginson Press. A more recent book is Cady Genealogy by James W Cady 1991 which I found with a site like Bookfinder.com. Also Ted Pack has put many families on his web site, do a search. Lastly. see http://trees.ancestry.com/tree/13444869/family?cfpid=3147014...
Herindeen Family Tree (Pay Site)
Census of 1850 Hampton, Windham, CT shows Elijah and Martha with John A Cady age 5 (Abt 1845)
Census of 1860 Hampton, Windham, CT shows Elijah and Martha with John A Cady age 15 (Abt 1845)
CENSUS OF 1870 Hampton, Windham, CT shows Elijah and Martha NO JOHN
CENSUS OF 1870 Medford, Middlesex, Mass or Charleston, MA for John
A death Notice shows he died on 21 Sep 1875 in Boston, He was born in Hampton, CT. abt 1845 of Elijah and Martha
Louise Remarried 14 Feb 1878 Herny Bent (1844 Me) He appeared on 1850 Census as a Pauper in Monroe, Waldo, Maine
CENSUS OF 1880 Hampton, Windham, CT shows Elijah and Martha NO OTHERS
CENSUS OF 1880 Medford, Middlesex, Mass Shows Louise Bent remarried to Henry A Bent (45)
A Death Notice shows Martha Adams Cady died 9 Mar 1884 in Hampton, CT
A Death Notice shows Elijah died 29 Dec 1888 in Hampton, CT Also, Austin A Cady age 20 Emma Cady age 23, Horace Cady age 25
Census of 1900 Willimantic, Windham, CT Harry, Margaret and Harry Jr He shows up on all later Census'
Suggestion: The Male line is often easier to follow. But, for you the Female line might be more important in the future, MtDNA or Mitochondrial DNA. Every mother passes this on to her Children. Brian Sykes wrote the Seven Daughters of Eve about European Eve's, Knowing which "Clan" you are tied to and when they arrived in America, can be interesting.
Obviously, Harry had some difficulty in life losing his parents and having a step-father who was himself a foster child, I'd bet his wife was critical to their children's success.