Linda, the three Price's who served in Co. H, 35th Arkansas, with Edmond Guthrie were Christopher C. Price, James F. Price, and Reuben C. Price.
I'm not really sure what prompted the rise in desertions in May 1863. There were two more peaks in desertion, in July and August 1863 (following the debacle at Helena), and in September 1863 (following the fall of Little Rock). But what the significance of May 1863 is, I can't put my finger on.