Both sides had understrength units. New units were raised by various states, and made available to the central governments. When soldiers were wounded, the regiment got along without them. When they died, the regiment got along without them. New regiments were raised. Occasionally an old regiment would be disbanded, or upon death of the commander could be combined with another regiment from the same state. New commanders would come with new men from the state, being politically appointed by the Governor. Many of the younger officers would be elected by the men in their company.
Sherman wrote that new units were not as useful as the old units, and that a replacement system would be better to put new men under experienced officers and sergeants. He wrote, after the war that nearly all methods of raising men had been tried, conscription, bought substitutes (mercenaries), and volunteers. Of those, the volunteers were the best.
I read a story about some Texas men from Hood’s brigade that went to Jeff Davis to protest the planned merging of their unit with men from other states. Don’t blame them, just that all units at that time were, after their first battle, understrength, and there was no systematic means to push reinforcements to depleted units. Wounded men sent home to get well in Georgia in particular were redrafted by their state, and sent off in new units, rather than being permitted to return to their old unit. A soldier could be considered AWOL from his old unit while serving in their new unit, and if caught in that situation could have some ‘splaining to do.
The individual replacement system used by the US during WWII was also a problem. New men were often pushed forward to depleted units with little time to train with and integrate into their receiving units. Some died before anyone even knew their name.
German practice tried to pull units off the line so replacements could train with experienced cadre, but as they were pressed very hard at the end of the war, the cadres were increasingly small, and the time available for training decreased.
I am not sure that anyone has the right answer on personnel management in war, even today.
The fact that a ideally an army corp was supposed to be 30,000 men strong (3 - 10,000 man Division) and the Feds fielded 9 corps tells you some thing. A Corp in Meade’s Army was really a division, you go down from there. So usual you are full of Lincoln.