So, for better or for worse, I am researching all the songs in the Rise Up Singing Songbook. Most of the songs in the book are already here, but this one isn't. I learned it in Cub Scout leader training here in Northern California in the mid-1980s. It appeared in Volume 32, No. 2, of Sing Out! Magazine in the Fall of 1986, page 43:

PEANUT BUTTER AND JELLY

CHORUS:
PEANUT, PEANUT BUTTER...AND JELLY, AND JELLY (twice)

1) First you take the peanuts and you pick 'em - and you pick 'em
[Bend to the ground and pick peanuts.]
Then you take the peanuts and you shell 'em - and you shell 'em [Stand straight and shell peanuts into an imaginary bowl]


2) Then you take the peanuts and you chop 'em - and you chop 'em
[Left hand holds chopping bowl - right hand uses chopper up and down]
Then you take the peanuts and you moosh 'em - and you moosh 'em
[Left hand holds chopping bowl - right hand mooshes up peanuts in bowl — going round and round in stirring motion]

3) Then you take the grapes and you pick 'em - and you pick 'em
[Reach high on the vine and pick the grapes with both arms]
Then you take the grapes and you moosh 'em - and you moosh 'em
[Hold the bowl with your left hand. Stir and moosh grapes in the bowl with right hand going round and round]

4) Then you take the bread and you spread it - and you spread it
[Hold out sour left palm with the fingers together. Spread peanut butter and jelly with an imaginary knife in your right hand.]
Then you take the sandwich and you eat it - and you eat it
[With both hands eat the sandwich with gusto!]

Final Chorus: [Sing one time as if sour mouth is all gooed up with peanut butter. Try to clean your teeth with your finger as you sing.]



Source: Sing Out! Magazine, Vol. 32, No. 2, Fall 1986, page 43


The Sing Out! transcription has the chorus "PEANUTS, PEANUT BUTTER...AND JELLY, AND JELLY," but I think they're just wrong.

Here's a YouTube recording of the song by Cathy Fink and Marcy Marxer:

https://www.youtube.com/watch?v=cxb4WTcl9T8

I sing it more-or-less the same, but tend to sing "PEANUT, PEANUT BUTTER" at annoyingly loud volume, and whisper "AND JELLY, AND JELLY."