I strongly suspect that it was honeysuckle, since that’s exactly what we did when I was a child in Louisiana (KG in '78). You can check that link for a pic of Japanese honeysuckle to confirm it.
We didn’t do anything with dandelions but blow the seeds away.
Holly, what you’ve got there is an appropriation of lyrics from an old song. I’m not sure the date of the song, but I’m guessing that it dates from around WWI. It goes:
====
Playmate, come out and play with me
And bring your dollies three
Climb up my apple tree
Hide in my rain barrel
Slide down my cellar door
And we’ll be jolly friends forever more.
She couldn’t come out and play
It wasn’t a rainy day
With tears in her eyes
She heaved a sigh
And I could hear her say
I’m sorry playmate, I cannot play with you
My dolly’s got the flu
Boo hoo hoo hoo hoo hoo
Can’t hide in your rain barrel
Or slide down your cellar door
But we’ll be jolly friends forever more
===
It also has a nice melody; nice enough that I have a music box which plays it.
Bingo! That’s the stuff…
Sparteye, good detective work! Words and music by Saxie Dowell, Copyright 1940 by Santly-Joy-Select Inc. I had completely forgotten about it until it until Kinsey and Holly brought it up!
Our version, Central Vermont, about 1979:
Oh, Mr. playmate,
Come out and play with me
climb up my apple tree
Slide down my rainbow
Into my cellar door,
and we’ll be jolly friends
forever more
Oh, little playmate,
I cannot play with you
my doggie has the flu
Oh boo hoo hoo
so climb your apple tree
out by your cellar door
'cause we’ll be jolly friends
forever more!
Oddly creepy, now that I remember it 
We also used to play a lot of handclap games on the playground. It was always just the girls, though. Anything that was a jumprope rhyme was used as a handclap game, or was modified so that it could be. The Cinderella ones, in particular, Miss Marry Mack, Miss Susie, and ‘Categories’.
We used the Cinderella/snake rhyme, and also the Cinderella/girdle rhyme:
Cinderella
dressed in yella
went downtown
to see her fella
on the way her girdle busted
how many people were disgusted
1,2,3…
We, also used to do the ‘mama had a baby’ thing. Everyone knew it, we would play at it for a while as a fun contest to see who could shoot the ‘head’ the farthest before we settled down to making sticky dandelion flower chains to wear.
I strongly suspect the “Playmate” song is well over a century old. I first heard a version of this from the Warren Beatty film Reds. Tune’s different, of course, but the chorus is strikingly familiar.
Just a bit of folklore to throw into the thread.
bughunter, you gotta show me how to do the knotted stem-gun thing. I could never get any distance from just flicking it with my thumb.
I’ve had that damn “Say, say, oh playmate” song stuck in my head for days now.
I find myself singing it at all hours of the day and night.
I’m going to go starking raving mad!
Help!!!
[sub]Say, say, oh playate, come out and play with me, and bring your dollies three…[/sub]
Aaagggghhhhhhhh!