The National Archives digitization program for Confederate Record Group 109 is available on line.

The postal route volumes can be found here:
https://catalog.archives.gov/search?q=*:*&f.ancestorNaIds=438&sort=titleSort%20asc&f.fileFormat=application%2Fpdf&f.level=fileunit&tabType=online&rows=100&offset=1100

This is the sort of information provided, but I did not see anything about the postmasters.

Post Office Department - Records of Post Offices, Texas and Arkansas
Chapter XI Volume 50

office counties Routes from which mail is supplied
94.06 Fort Washita Cherokee Nation 7948, 7949, 7952

Post Office Department - Route Book, Arkansas, 1862-1863
Chapter XI Volume 17
410 [7952A] From Doaksville by Armstrong Academy to Fort Washita
90 miles and back, once a week
Contractor: David W Heard, Fort Smith
$1250 accepted Sept 26, 62
D C Betts of Doaksville by letter of July 31, 63 rec’d June 10, 64 says by agreement with W Watson, Heard’s agent, he carred fpr 12th of Jan 63 up to 31st July 63

Schedule
From Doaksville Mon at 5 am
Arrive Fort Washita next day by 6 pm
Leave Fort Washita Wed at 5 am
Arrive at Doaksville next day by 6 pm

407 [7949] From Fort Washita by Tishomingo to Fort Arbuckle
70 miles and back once a week
Contractor: David W Heard, Fort Smith
$1100 accepted Sept 26, 62 [failed to organize]

See No 179 Texas RR

The service having been abandoned, instructions were on June 11, 64 given to the [IRw?] J C Robinson to advertise the route for service I.E.aw.; that …