I’ve heard the expression “my dance card is pretty full” and similar and I’ve sussed that a full dance card implies being busy, but what is a dance card, what is it full of, and what are you busy doing when it is full? Dancing?
In genteel society of past, a gentleman would never dream of just walking up to a lady to ask her to dance, nor would the lady accept.
Her dance card was like an appointment book. Once an affair was planned, a gentleman interested in a particular lady, would call at her home prior to the event and sign her dance card. It was impolite to call the day of the affair. That would imply the lady was less than attractive, or popular. Having a full dance card was the the social equivalent of being a regular at Studio 54
The link is the dance card museum. The dance card was physically something the lady cherished. Most were very ornate. She kept the same one from her coming-out until she married. Then it was tucked away as a memento of her lost youth.
In the business world, a dance card is your schedule of meetings for the day, it’s very “corporate speak”.
I believe that dance cards survived through World War II when dancers working for the USO would use them to work a room full of soldiers. “Sorry, my dance card is full.” was an easy way to give the brush off to someone a dancer didn’t like.
ccwaterback, I’ve heard plumbers use the term as well.
Dance cards were still in use for many non-school dances when I was in my mid-teens in the early 60s.
Like picunurse said, it is a card on which a gentleman would write his name to reserve a specific dance with a lady. My great-grandmother apparently was quite social, and my grandmother still has some of her dance cards. They are all small – about the size of the palm of a man’s hand – and had a ribbon threaded through the corner. My grandmother said that was to tie the card to the lady’s wrist.
I don’t doubt it. I think the only guy I heard use the term often was a “Mr. Corp. Speak” a happened to work with a few years ago. So, anytime I hear “dance card” or “drill down to the details”, etc. it reminds me of that hapless fellow.
I believe they were used by Dime-A-Dance well beyond polite the time when society bothered with them. If you make your money on tips gentlemen give you in exchange for manual release, you’d better know exactly who’s next.
For the life of me, I’m trying to understand these two sentences and am failing miserably. 
When someone asked you to “save me the waltz,” you knew they were really serious about you.
Then you had to learn The Language of the Fan.
“Dime-a-dance” halls were establishments that ran the gamut from seedy almost-whorehouses to fairly high-class establishments (though on the high end a dance almost certainly cost more than a dime). Think Sweet Charity or Tina Turner’s “Private Dancer” or (on the incredibly silly side) Pat Benatar’s video for “Love Is A Battlefield”.
First, the quintessential dime a dance song is Ruth Etting’s rendition of “Ten Cents a Dance.”
Second, I will brook no dissing of the “Love is a Battlefield” video!
Thanks! (along the lines of what I suspected, but some awkward sentence construction was really getting in the way) 
It wasn’t the sentence construction. I just had a couple words try to escape, there. They nearly got away.