He seems to be in Talbot Georgia in 1860. The census record shows wife Missouri and Children;
John 12,Thomas 10,Mary 8, Daniel 4,Sarah 1. All born in Georgia. It gives his birth year as 1825,Georgia.
"United States Census, 1860", database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:MZMB-WZB : 18 February 2021), Adin Hamilton, 1860.
He married Missouri Bane Dec.3,1848,Crawford Georgia.
"Georgia Marriages, 1808-1967", database, FamilySearch (https://familysearch.org/ark:/61903/1:1:FW3P-1HQ : 11 January 2020), Aden Hamilton, 1848.
Like you said he was in the Civil war.28th Battalion, Georgia Siege Artillary. Private Company D
"United States Civil War Soldiers Index, 1861-1865", database, FamilySearch (https://www.familysearch.org/ark:/61903/1:1:FS7P-62R : 25 August 2021), Aden Hamilton, 1861-1865.
According to FAG His military grave marker may be in Geneva,Sabine,Texas Death Date 1864. Myrtle Springs Cemetery.
"Find A Grave Index," database, FamilySearch (https://www.familysearch.org/ark:/61903/1:1:QVLD-2FN5 : 24 February 2022), Adin J. Hamilton, ; Burial, Geneva, Sabine, Texas, United States of America, Myrtle Springs Cemetery; citing record ID 93946041, Find a Grave, http://www.findagrave.com.
A record on Fold 3 states he was in the hospital in Charlotte N.C. Feb6,1865 , but sent to another hospital. Does not say where. I am assuming Texas, since the FAG records says he is buried there.
I hope this helps!