Sweetums
I think the mouth of Sweetums was performed using the performers right hand, which could go into the right hand. there are two scenes with sweetums that I think makes this obvious.
At the beginning of the great muppet caper, when sweetums is running and falls into the sewer, he talks but his mouth doesn't move. both his hands move though.
In the john cleese episode, during the "we got each other" number,it looks like his right arm hangs down and drags, but in the middle of the number, he starst dancing and moving both arms, but his mouth is closed the whole time. His right arm starts moving a few seconds after he shuts his mouth, and in the same take, so I guess it is easy for the performer to switch the arm from being in the mouth to being in the other arm.