Is it correct to say that a barkeep was "adroitly pouring out drinks behind the bar"? Something rubs me the wrong way here. Perhaps it would be better to use some synonym in this context? If so, which one?
It is absolutely grammatically correct. I suspect that it is the fact that adroitly is very formal and is not commonly used in everyday speech that is making it sound incorrect to you. To be honest, if I were speaking about a bartender, I wouldn't use the word adroitly, but that's simply my personal preference; that doesn't make it incorrect.