what does "down a stiff shot" mean? drinking heavily?
Dave Hackett, who had known Bob Kennedy since boyhood, couldn't bear to be among friends; he left a meeting and scoured the neighborhood for a taproom where he would be unknown and where he could down a stiff shot unnoticed.

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In this context, "to down" means "to drink" and "stiff" means "strong". So, "a stiff shot" is "a strong shot (of alcohol)".

"to down a stiff shot" means "to drink a strong shot (of alcohol)".

But it doesn't necessarily mean he drank heavily. He may have only had one shot. But if he drank three or four shots, then yes, he would be drinking heavily.

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A stiff shot means a drink with a high alcohol content.  He went to a taproom (bar) and had a stiff shot (a strong alcoholic beverage).

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He wanted to drink (down)  a small glass of liquor (stiff shot).  It's not that he wanted to drink a lot, just that he could get relief with a shot of braday, whiskey, or the like.