As we suspected, the familiar part of this song is only the chorus, and there are two seldom-sung verses. I transcribed this from the sheet music at the Levy site that MMario linked to above. Interesting to note that in some of the remembered versions quoted above, the genders are reversed.

SCHOOL DAYS
WHEN WE WERE A COUPLE OF KIDS
By Cobb & Edwards, 1907.
Valse Moderato

Nothing to do, Nellie Darling, nothing to do, you say.
Let's take a trip on memory's ship back to the bygone days.
Sail to the old village schoolhouse. Anchor outside the school door.
Look in and see. There's you and there's me, a couple of kids once more.

CHO: School days, school days, dear old golden rule days.
Readin' and 'ritin' and 'rithmetic,
Taught to the tune of a hick'ry stick.
You were my queen in calico.
I was your bashful barefoot beau.
And you wrote on my slate, "I love you, Joe,"
When we were a couple of kids.

'Member the hill, Nellie Darling, and the oak tree that grew on its brow?
They've built forty stories upon that old hill and the oak's an old chestnut now.
'Member the meadows so green, dear, so fragrant with clover and maize?
Into new city lots and preferred business plots, they've cut them up since those days. CHO.