Have you copies of Oklahoma marriage certificate and father's birth certificate? Do you know a middle initial? Where born and age. One or both should be on those records.
There are several Carl Edwards that enlisted...
There is a Carl L. Edwards in the database "U.S. Rosters of World War II Dead, 1939-1945" but it does not have a death date. He was buried in the Margraten Cemetery, Holland (per next of kin). Has he was an Army Private with Serial Number 39477250.
At the American Battle Monuments Commission website it is known as the Netherlands American Cemetery. Just now tried to do a search of burials but it is "down". The search engine reports no results in that cemetery for last name Edwards. But it also has no results for any surname I tried in any cemetery for any war. You might try later, as it usually give Unit and place enlisted from.
http://www.abmc.gov/search-abmc-burials-and-memorializationsNow the NARA has online U.S. Army Enlistments at the ADD Website and it has Service Numbers (Ancestry's same database does not.) It seems a Carl L Edwards with the above service number, enlisted in in Ft Lewis Washington and was from Multnomah County Oregon. Age 19 married, enlisted 1944.
http://aad.archives.gov/aad/record-detail.jsp?dt=893&mtc...There is also a Carl M Edwards that home is Osage County OK, b. 1927 enlisted in Oklahoma City, 1945. Was married and Army Service Number 38781018. Ponca City is in both Osage and Kay Counties.
http://aad.archives.gov/aad/record-detail.jsp?dt=893&mtc...Divorce was still somewhat of a "shame" back then. He may have come back and wanted a divorce or stayed overseas and wanted a divorce so that is why finding out what status your grandmother claimed with second marriage is important. As well as he age and place of birth. Also she may have gotten the divorce......
Edit
SSDI has a Carl M Edwards b. 1927 SSN issued in Oklahoma d. 1983 last address Wichita, Sedgwick County Kansas.
An obit or death notice is in the Wichita Eagle and Beacon, can get copy through Historical Society:
http://mhgswichita.org/83obits.html There are only 4 men named Carl Edwards in WWII casualties lists...
Carl A Edwards. Navy ...missing at sea
Carl L Edwards. Army...buried in Netherlands, information above
Carl L Edwards Army ...buried Iowa
Carl W Edwards Army...buried TX. Carl Weldon Edwards in NARA ADD