Possible Clues:

• Seems you get an odd sort of a degree from South America
• What needs some decorum - ballroom dance
• That's a peculiar sort of 27 across for the dancers
• Game preceding business administrator's dance
• Dance that's unconventional to a degree!
• Drink sailor knocked back after dance
• Eccentric person of education has Latin beat
• Dance is sweet, though a degree short
• Arthur Murray lesson
• Latin dance? Curious dance, stopped halfway through