I have puppeteered a character similar to Sweetums which could be configured a couple of ways.
The mouth was "Sprung" to remain in a naturally closed position. My hand was in the lower jaw only, there was no control for the upper. It was almost a slapping action to activate the mouth.
The eyes were actually controlled with my chin on a control bar. Rather ingenious, but not entirely efficient!
The right hand mostly just hung at the side, but when it needed to operate I could easily switch my hand from the mouth to the hand for manipulation.This is very much what Tztz suggests, although I didn't need a separate setup to switch from hand to mouth, I could do both in one take, just not simultaneously.
As far as "eating" goes the key here, regardless the puppet, is just getting out of the way! I have used hand puppets that "eat" and I just slid my hand as far to the side as I could to make room the the "consumables". In "Little Shop" Twoey obviously does a lot of eating...in the #3 puppet eating Orin's entrail's consisted of an open throat and a puppet "tongue" to assist in shoving the items down into my lap. In the #4 puppet Twoey eats Mushnik, Audrey, and Seymour. Each was accomplished differently.
For Muschnik I would straddle as widely as I sould and lift the upper pod with my back. He would step into the pod and I would slam down over him. For the second chomp he would fall back and I would bite him firmly across the chest, in a little tribute to Robert Shaw in "Jaws". For the third chomp he would fall forward and dart out between my legs, out the back of the puppet. During this action I made many smaller "chewing" motions. Then we (voice actor and I) would do a big laugh, which was not only menacing but it showed that Mushnik had truly been "eaten' and was not merely hiding inside the pod.
For Audrey it was similar but much slower. I would raise the pod and Seymour placed Audrey within. As I very SLOWLY closed the pod Audrey would reach up and grab the control bar with one hand, and my wrist with the other. Then together we would slowly pull her feet through the "lips" of the pod while the voice actor provided a very nice spaghetti slurp sound.
For Seymour I would stand sideways in the pod and lean as far downstage as I could. Then when Seymour ran and leapt into the pod I would slam and "chew" from the front of the pod while Seymour escaped out the back. I would then get the machete (that Seymour dropped in the front of the pod), move to the rear, open the mouth tossing out the machete, to another well-timed "spit" sound from the voice actor.
So, like so many things, each puppetry challenge can be unique and require equally unique solutions. If we hadn't devised 3 different methods the audience would have grown to expect the same treatment each time. By varying the method it increased the "how'd they do that?" factor.