Unfortunately, I think the Adam I have information on is a different Adam.Below is a brief biography:
In January, 1864, Adam Cosner enlisted in the 21st Ohio Infantry at the relatively advanced age of 43, leaving behind his wife, Ann, and children in Findlay. After only four months of duty in northern Georgia, marked most notably by his industrious efforts to make money by doing other soldiers' laundry, Cosner fell ill and was sent to hospital in Chattanooga where he remained until August. Early in September, he returned to his regiment, and served during the later stages of the Atlanta Campaign and on Sherman's March to the Sea.
Adam Cosner's letters reflect his rather unusual position as a middle-aged soldier. Sixteen of the 18 letters were written to his wife, Ann B. Cosner, and the other two were addressed to his daughter, Martha, one in 1880. The letters indicate that he was not among the more educated soldiers in the Union Army. His handwriting suggests that he was not an experienced letter-writer, nor are his descriptive skills well developed, seldom advancing beyond straightforward, unreflective comments and never at length.
Adam is buried in Woodlawn Cemetery in Toledo Ohio.He was a son of Philip Cosner & Dorthea Coler of Knox County Ohio.
I'm sorry that this is not the Adam Cosner you are looking for.