The song was a hit by the Chordettes, yes. In 1959.
It has nothing to do with Jefferson Airplane, no.
In 1959 it was also done by The BROWNS (well known by their mega-hit The Three Bells).
In Germany it was a hit by Lolita (title "Treu will ich dir bleiben", or something like that). Also 1959.

Lyrics:
WE SHOULD BE TOGETHER as sung by THE BELLS
(John D. Loudermilk)
© '59 Acuff-Rose Music

(Why why why can't we be together why why why can't we be together)
We should be together like the rivers that are aiming to the ocean
We should be together like the clouds that are floating in the heaven
Melted together like sand and the sea rocks and the earth and wind and the breeze
Darling everything's together but you and me
(Why why why can't we be together why why why can't we be together)

We should be together like the gold that veins in the mountain
We should be together like the grass that grows in the valley
Melted together like the surf and the sea
And new fallen snow and the leaves from the trees
Darling everything's together but you and me
(Why why why can't we be together tell me why why why can't we be together)

Love is the only thing neath the sun that can make two lovers into one
Darling I'll be waiting until it's done
(Why why why can't we be together why why why can't we be together)