You really need to listen to the original (most of the laughs are in the timing and vocal intonations), but for those of you who haven't yet, here is a reasonably good feel for the material
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Transcript from http://home1.gte.net/longrj2/fluff/elderlymanriver.html

I hope to have a transcript of the entire recording not long from now. (Need to find which cassette tape it's on.) For now, here is most of it:

The setting is as follows: A musical radio program is going on and Stan is the host. Just before he introduces the next song, a man walks up and announces himself.

Man: My name is Tweedly. [he places emphasis on, and draws out, "Tweedly".]
Stan: Well, we all have our problems.

Tweedly states that he is there to ensure that Stan does not play anything that he finds offensive. Tweedly will activate his horn every time Stan does something offensive. Stan is hesitant, but decides to go on and starts his introduction.

Bzzzzzt! Stan forgot to be polite and thank Mr. Tweedly.
Stan: Why, that's a darling little horn you have there, Mr. Tweedly.

Stan introduces the next song, "Old Man River", and begins to sing.
"Old man river, that old man..." Bzzzzzt!
Stan: Okay Tweedly, politeness I dig, but what is wrong with "old man river"?
Tweedly: The word "old" has a connotation that some of the more elderly people find offensive. I suggest you make the substitution.
Stan: I suppose you insist.
Tweedly: Naturally.

Stan begins again.
"Elderly man river, that elderly man river. He must know somethin', but he don't say nothin'." Bzzzzzt!
Tweedly chides Stan for improper grammar.
Stan: "Somethin'," "somethin'," it's authentic.
Tweedly says that proper grammar should be used at all times, since "we must be a good influence on... children." [There's a pause and then 'children' is drawn out.]

By now, Stan is starting to get the hang of this, and begins again.
"Elderly man river, that elderly man river. He must know something, but he doesn't say anything. He just keeps rollin' [catches himself] rolling, he just keeps rolling along."
"He don't [Bzzt] doesn't plant taters [catches himself] potatoes,
he doesn't plant cotton [catches himself] cotting, because these/them/those that plants them are soon forgotting.
Elderly man river, he just keeps rolling along."

The song continues...
"...Body's all aching and wracked with pain." Well, we got by with that one.
"Tote that barge, lift that bail. You get a little..." [Stan slows down and stops here, since the rest of it is "drunk and land in jail".]
Stan: Okay, Tweedly, you can take your finger off the buzzer.

Stan gives up at this point, and wraps up the show.
Bzzzt
Stan: Oh, yes, and thank you, Mr. Tweedly.
Tweedly: You're quite welcome, I'm sure.

~~~~~~~~~~~~~~

See also Wiki Ol'_Man_River

A parody version was performed on CBS Radio by Stan Freberg and Daws Butler circa 1960, entitled "Elderly Man River." The parody lampooned what would today be termed "political correctness" by featuring a prudish censor from the "Citizen's Radio Board" who repeatedly interrupts Freberg's performance of the song to criticize (and insist on changes on) the grammar and appropriateness of the song's lyrics.

~~~~~

The track is still available today on CD, one of the True Comic Classics.