I agree it all seems to fit, I did a quick Ancestry.com census 1860 Desoto Parish search, and find only one Daniel Adams, city of Orleans, La., age 35 in 1860.
I do not have a subscription, hence, I can search no further.

However, I find it difficult to believe a 34 year old, married Farmer, with children most likely, re-up having already been discharged for physical reasons. He would have been clearly exempt from further service.

Re burial, had he died in camp, vs. a hospital, his comrades would have buried him nearby, in a cemetery if close by. They would have marked his grave with a simple wooden marker. These markers would have disintegrated over time, or perhaps have been used for firewood, lost, etc.

Unfortunately, he lies at an unknown site as thousands of other casualties of war.

You can still obtain a VA memorial tombstone and set in your family cemetery if desired.