Possible Clues:

• Seems like an odd sort of degree you get from South America
• Odd sort of drink to a degree at the end of 28 across
• Lively Cuban dance
• Funny start to a barn dance
• Dance and drink on the airline
• Dance tune that's peculiar with band half cut
• Afro-Cuban or ballroom dance
• Dance fragment: missing the start, but with a conclusion
• A measure of drink to the graduate from South America
• Gets drunk to an odd degree from South America