I've heard about Frank putting his hand on the right-hand performers hands so they wouldn't gesture too much. Though I don't know of many segments where it's obvious his character's holding onto the left hand (I think I've noticed it in a few Harvey Kneeslapper segments).
I think the hardest thing about being a puppeteer is right-handing; the 40 Years book really delved into that and just what the real challenges were that went into it: I read Tyler Bunch said being a big person has prevented him from doing a lot of right-handing (once right-handing for Baby Bear, and during a break, David said in Baby Bear's voice, "I've never been so aware of my wight hand befowe in my life"), so he ends up with a lot of minor one-shot characters. It's ironic, because some text has said that the right-hand is the hand that does most of the action, but speaking with right-handers they pretty much say the opposite... as with Frank, I think that's probably his way of reminding them you're not the performer here, I am, I'll lead, you'll follow.
I wonder if that might explain him being better as Big Bird and Oscar, since those two were primarily on the street while Jim didn't perform on the street often, allowing Caroll to concentrate better without the awe of performing with Jim.
Perhaps, but to his credit, I can't exactly say that his performances as Big Bird and Oscar were exactly stellar in those early days either, but then again, that was partly due to poor puppeteering conditions: the first Oscar was built specifically for a right-hand, but the way they had Oscar's home set up forced Caroll to work the puppet left-handed, which pretty much ruined the puppet... not to mention, the first Big Bird didn't have a monitor inside it, so he really couldn't see much of what he was doing... as a puppeteer, I can attest to that, the monitors are a HUGE reliance, without those, you're practically working blind, and it's a tremendous handicap.