"Vuole la mia bevanda?"
I just mistakenly answered "Do you want my beverage?" and went all smoothly. But my answer was wrong! The fix is needed.
This should be a pleasant surprise: "formal you" is conjugated like he and she. You'll learn more about it further down the tree!
"Would you like my drink" was my answer, and that is how I would put it in English. At any pub, holding out my glass towards someone with a quizzical face.
I agree that while the direct translation is "Do you/Does she want my drink?" most English and American speakers would say "Would you like" - so the hint to not mistake "want and like" is really incorrect.