There could be some difference in how they store and preserve the puppets that affect how they look on camera. I was on the Sesame Street set in 2010 when they filmed "The Very End of X" episode. It makes a difference, I think, depending on what character they are getting ready to film.
Abby wasn't in that episode, but I did see her on one of those poles near, but not touching any other puppets on sort of a hatrack-type thing. My guess is that they had to keep her pristine because of her pigtails and the sparkly...um...(glitter?) in the ostrich feathers. (It's hard to see what exactly was in there. Even in HD, the camera can't pick up what you see in real life).
They had a Fat Blue (as a cowboy) on the table, but sitting upright and on one of those poles. Again, he was a little ways away from the other characters, but the Sesame Workshop folks didn't seem to mind his pants touching the table.
There was an ox, which was furry, but they had just, well, tossed him on the table. It didn't seem to matter that his fur got a little matted (or ruffled up) for the shoot.
We all know how Snuffy is stored. Still, they were careful on the set to brush him by hand when Marty and Bryant Young walked him to the arbor. The puppet wranglers looked like they were petting him, actually. It's reddish-brown ostrich feather strands, so they did kind of float a little. (One fell on Leela's face just before they shot her scene).
Big Bird, as we know, was handled by white-gloved folks and Caroll together when he got into the bird. The wranglers had to "pet" him, too.
As for Oscar, well, I saw Caroll perform Oscar on the set, even though he wasn't in that episode. (He entertained a few fans). He might look clean on the screen, but I know exactly how Caroll stores him. Caroll takes Oscar off, rolls and wads him up and throws him in a puppet cabinet in the corner. Yeah, just like how a kid would treat an itchy sweater they got from their Aunt Gertrude.
I'll have to find it, but there's something Kevin Clash said about how the "used look" of the characters (like having a baseball glove that fits) looks good on camera.
So, maybe the pristine look might have to do with character or whatever materials they're made out of. I agree they need to improve Fozzie's look, though. Why doesn't Puppet Heap hire Terry Angus, for crying out loud?