It helps me to review the service files for the men you mentioned. First, John R. W. Lannom doesn't appear to have a service file. If service could not be confirmed to the satisfaction of the pension board, his claim would have to be denied.

Second, you mentioned Nathan P. Lannom, Co. "G", 7th Tennessee Regiment, Army of Northern Virginia. After being wounded at Cedar Mountain, Va., Aug. 8, 1862, he was sent home on furlough. If a soldier was disabled by wounds, he would have been discharged. It would not serve any purpose to give a soldier an unlimited furlough. At any rate, NPL never returned and was reported as a deserter. The record shows he was captured at home by Indiana cavalrymen, sent to prison, and returned on exchange. I find no record that NPL ever served in another command.

I don't see an oath of allegiance listed for either man.

Maybe you should try someone else. There are several Lannoms in the 7th Tennessee Regiment (not cavalry).