Gooday Chris,
Both Alexandria & Dyersburg where transferred to Third Air Force from Second Air Force early in 1945 and used the alpha/numeric codes of Third AF for III Bomber Command. Y for Dyersburg and X for Alexandria. Both are crew replacement training bases.
Both Williams and Roswell are pilot transition schools of AAF Western Flying Training Command note the lack of guns on these transition ships. Y for Williams and W for Roswell with the last three numbers of the serial number on the tail. I think that both Williams & Roswell are assigned to the same Flying Training Wing of Western Flying Training Command to have this system.
A lot of Second AF bases used a two letter field disignation and last three of the serial number on the tail, like RC for Rapid City as shown, Alexandria used AL and Dyersburg used DY when assigned to Second AF.
They are lots of duplicate field letters for HQ AAF left that up to the various reporting commands and air forces to do that.
Here is a B-17 transition trainer at Williams being welcome back to base from Life magazine:

All the best
Tom