Dear Ms. Wills,
We searched the National Archives Catalog and located the series Carded Records Showing Military Service of Soldiers Who Fought in Volunteer Organizations During the American Civil War in the Records of the Adjutant General's Office, 1762 - 1917 (Record Group 94) and located 3 service records for persons serving in West Virginia units on the Union side. These records have been digitized and may be viewed online using the Catalog. We also located 2 records relating to Floyd Cook.
We searched the National Archives Catalog and located Carded Records Showing Military Service of Soldiers Who Fought in Confederate Organizations , 1903 - 1927 in the War Department Collection of Confederate Records (Record Group 109) that includes 16 records for soldiers from Virginia named George Cook, which may include persons that had lived in what became West Virginia. These records have been digitized and may be viewed online using the Catalog.
In addition to the service records, there may be pensions records for the Union veterans which may provide more information. To order Union pension records, please fill out NATF Form 85 for pensions and mail the completed form to the address listed on the form. For more information see Requesting Copies of Older (pre-WWI) Military Service Records.
Please note that Confederate pension records are in the custody of the various state archives, not at the National Archives.
We also searched the National Park Service Civil War Soldiers and Sailors System (CWSS) and located similar results for the name “Cook, George” for West Virginia and Virginia. We were unable to locate a listing for Floyd Cook in the CWSS.
West Virginia Archives and History may have additional information about soldiers from West Virginia who fought during the Civil War.
We hope this information is helpful. Best of luck with your research!