The hospital record date/location you found (5 July 1864, Marietta, GA) jibes with the known location of the 37th Alabama at that time, although it's hard to know exactly when his wound may have occurred "a few days" prior to his hospitalization.
The regiment was engaged at "Mt. Zion Church" (a.k.a. "Kolb's Farm") on 22 June, and again at Kennesaw Mountain (24-27 June).
There's an excerpt from a first-hand account that I've put on my site, written by Pvt Benjamin F. McPherson, Co I, 37th Ala, about what he experienced at Kennesaw Mtn that may give you some insight into the conditions Sgt Wright was in and around when he was wounded (or endured after he was wounded), along the line there. The passage is located at <http://alabama37th.com/History-h.htm>.
I'll update Sgt Wright's bio on my site with the Alabama Archives data you provided.