Confederate Brigadier General John Vaughn had a whole three-regiment brigade of conscripted East Tennesseans that broke and ran at Big Black River Bridge during the Vicksburg Campaign; those that weren't captured (after putting up feeble or no resistance) were considered untrustworthy and assigned to a quiet sector of the Vicksburg defenses for the duration of the siege. This wasn't a unique problem, either for the reluctant conscripts or the Confederate authorities, especially as the war ground on.
As for Mosby, he was a realist and felt that the South would get a better deal if they acted reasonably and accepted defeat and the new Federal laws. He hadn't favored secession in the first place, but went with his state for everything he was worth, never relenting until the war was unquestionably lost and over with. It really wasn't until after the death of his wife that he began to actively advocate Republican political viewpoints, angering his friends and neighbors and turning him into an exile from his native state.