If you were going to study three Civil War governors, they would have to be Dennison, Morton, and Yates. Their impact on the war was boderline Cabinet level, with Dennison eventually joining the Cabinet. They were exceptional men and loyal Lincoln political operators whose common denominator was personal (career) indebtedness to the President.
Republican General McDowell (of Bull Run) was Dennison's cousin; Dennison seems to have appointed McClellan at Lincoln's behest (the evidence is circumstantial); and Lincoln had him convey to McClellan his relief from the position of general-in-chief (Mac was loading troops, missed his appointment, and read of the relief in the newspaper).
Dennison, Yates, and Morton were part of Lincoln's war management system, his regional (midwest) high command, and military patronage apparatus.
You can make a small start on Dennison here. Battle-centric publishing seem to preclude these men from getting any attention in the near future.
p.s. This is from the same Cincinnati Enquirer that drove U.S. Grant crazy during the Rebellion.