You beat time on my head With a palm caked hard by dirt, (lines 13-14)
We know that the time being beat is just for keeping time in the waltz, but this line also shows how children learn about time from their parents in general. They look up to their folks to tell them when to sleep and when to wake up, when to eat and when to play. In this case, we suspect that the time they get together before the father's death is slipping away.
Then waltzed me off to bed Still clinging to your shirt. (lines 15-16)
This little boy, like many little boys, doesn't want to go to bed. This shows that he's happy being up with his parents, no matter how rough this whiskey dance is. He loves his father so much that he tries to hang on to the man's clothing so that he won't leave him alone in bed.