Now I have the answers, and both answers are "yes" and "sort of". These answers come from a recent biograph of the Weavers, "Wasn't That a Time". This book documents that the WGN appearance captured here was their first TV appearance in years, and also indicates that, while in Chicago, they took part in a jam session at the then-brand-new Old Town School of Folk Music - the latter seemingly not a scheduled appearance in the evening after the TV program. This was certainly their last pre-arranged live performance.
What happened next was an studio attempt at making a rock-and-roll flavored single, which was panned by everyone involved, then a vote on whether or not to record a commercial for L & M cigarettes. Pete was outvoted, three to one, worked on the commercial with the group, then immediately quit.
That means this is undoubtedly the last recording of the original Weavers prior to the split. And there does not appear to be a copy of this performance anywhere online (it can be viewed at the Museum of Broadcast Communications Museum in Chicago, and I have seen it, but you can't take or get a copy).
A big additional benefit of this recording is the presence of the great Mahalia Jackson, who sings a few songs, including teaming up with the Weavers at the end of the hour long recording.
A huge deficit of this recording is the presence of the consistently and severely insufferable Richard Dyer-Bennett, a performer whose pretentiousness truly knew no bounds. I can imagine plenty of 1950's and 1960's people being unable to take folk music seriously, if they'd first been presented with Dyer-Bennett as a purveyor of the genre. He even manages to ruin "O No John", a fun, playful little number which was in my mother's repertoire. The Anti-Seeger, if you will. (I will add that the effect is even worse if you can see him in this performance which (as I've mentioned) I have.) Again, I'm sure there are those of you out there who's mileage varies. Such are the vagaries of taste.
But enough about him. Way too much.
This is probably the tape from my collection that I consider the most valuable. I would include the Merigail Moreland tapes and the raw tapes I own from an unreleased Dora Hall Dixieland album. The latter two, though would have a tiny, and a small audience interested, respectively. The audience for this one, I'm guessing, is quite large, and I've considered trying to sell it at times. But when it comes right down to it, I'm never parting with this tape, and if that's the case, I might as well share it with the world, or whatever part of the world is reading and listening to this.
Download: Various Artists - The Hour of Music, January 13, 1958, Part One
Download: Various Artists - The Hour of Music, January 13, 1958, Part Two
Happy Birthday, Pete! I miss you.