In recent years, it has been common for Big Bird to move his right arm more realisticlly. That's because in recent years it has been more common for Big Bird to either appear in front of a green background that get's replaced by a computer animated image or appear from the waist up (often by standing behind something). When these scenes are done, a second puppeteer performs the right hand (that puppeteer completely dresses in green if a green screen is involved).
When performing Sweetums, the puppeteer alternates between performing his right arm and performing his mouth. If Sweetums is moving his mouth, then his right arm holds still (either pinned to his waist or dragging), and if his right arm is moving then his mouth doesn't move, either being posed closed or open. Notice how at the beginning of The Great Muppet Caper right before he falls into the manhole, he is talking but his mouth doesn't move, yet both arms are moving.
In the John Cleese episode, during the Two Lost Souls number, at the beginning when Sweetums is singign his right arm doesn't move, but when Sweetums does a little dance in the middle of the song, after he closes his mouth his right arm starst moving a few seconds later (in the same take), and he doesn't open his mouht again untill after he stops moving his arm, which makes me believe that it is easy for a performer to switch from performing the mouth and the arm in a short period of time.