An Adjutant-General is the chief administrative officer of a military formation. Adjutant-generals take charge of personnel matters such as postings, transfers and assignments, the proper keeping of personnel records and the welfare and morale of servicemen. The term describes a job position and not a rank, so an adjutant-general need not necessarily be a general, although at a high level formation such as a Corps, it frequently is. The position of adjutant-general of a unit is normally held by an officer of the Adjutant-General's Corps.
In Season 8 "Too Many Cooks", Potter's friend Brigadier-General Budd Haggerty is an adjutant-general (probably of I Corps). In this position he would naturally know about Potter's request for transfer and so came down to the 4077th MASH to see why his old friend wanted out. As adjutant-general, he would also be in a position to change Pvt. Conway's MOS (military occupation specialty) from Infantryman to Cook.
Colonel Carmichael who presides at Hawkeye's hearing in Season 4 "The Novocaine Mutiny" is an officer of the Adjutant-General's Corps.
In Season 2, "Radar's Report", Frank and Margaret want a Section 8 for Klinger and threaten to take it directly to the Adjutant-General.
In Season 3 "Officer of the Day", Margaret is described as the Adjutant of the 4077th MASH, although later in "Big Mac", Frank says he is the Adjutant.
Colonel Carmichael presides at Hawkeye's hearing in "The Novocaine Mutiny". The insignia of the Adjutant-General's Corps is clearly seen on his lapels. He is also wearing the shoulder patch of the Eighth United States Army