Eakin's "Missouri Prisoners of War" (short version of long title) lists:
--Citizen Valentine Creger of Webster County arrested there 17 June 1862 and sent to Gratiot Street Prison in St. Louis, where he was released 19 January 1863 after taking an oath not to bear arms against the US.
All entries in her book were taken from the microfilm records of the National Archives and Records Administration.
Therefore, if somebody shot or hung him they did it after 19 January 1863. Since he was buried in his home county, my guess is that he was killed there, probably by the local Union militia there. Families sometimes had the remains of a loved one sent back home from a distant grave, but that was not done often in Union occupied Missouri with the mortal remains of southerners. One of the Webster County Union militias was the 74th Enrolled Missouri Militia, but to say they killed Criger is pure conjecture. Webster County was so close to the large Union garrison at Springfield that it could well have been somebody else. If you have his date of death or at least the month and year, it may help narrow the possibilities.
If I were to guess about David Cornelison's unit from the tombstone information, I would estimate he was a member of the 6th Missouri Cavalry Regiment (CSA), Company B. James McGhee's 2008 "A Guide to Missouri Confederate Units, 1861-1865" on page 77 tells us that the 6th was re-designated the 11th Missouri Cavalry Regiment led by Colonel John Trousdale Coffee of Dade County. This regiment(pages 93-97) was nicknamed "Southwest Cavalry," as it was composed mostly of men from southwest Missouri. On page 93 it says that Company B was mostly from Newton County and captained by John C. Toney. Does this sound like it fits David Cornelison, at least partially?