The jingle was kind of a Beach Boys knockoff. Sounded like "Surf City" to my middle-aged memory.
It's called Shake A Pudd'n
And all you do is shake it!
Shake, shake, shake a pudding
Pudding, pudding, SHAKE A PUDDING!
Now I wonder about the chemistry of Shake A Pudd'n, the way it set so quickly.
There were some related products. Remember GREAT SHAKES? Same shaker and tetrahedral packets but the end result did not gel as completely.
And after Cool Whip came out, there were two counterparts also in tubs. One was simply a tub of pudding, available in the usual vanilla, chocolate, and butterscotch. You kept it in the fridge. The other was kept in the freezer and created shakes. You took a couple scoops of the frozen concentrate, put them in milk, and stirred with a fork. Neat effect as the milk cooled.