"I am not cheerful, even though I am drinking wine."
Translation:Nie jestem wesoła, chociaż piję wino.
Yes, if there is a noun phrase coming after it. If it's just an adjective, it stays in Nominative. You can check here, Part 3: https://www.duolingo.com/comment/16373167
Not sure if this is also a part of the question, but "wesoły" will be what a man will say and "wesoła" what a woman will say. Adjectives need to match the gender, in this case the gender of the speaker.
This is a question that I find more and more often. The problem is, that you treat the construction of Present Continous too literally. If you had Present Simple, it would be clear: "drink" = "piję", no place for doubt. "am drinking" is just a grammatical construction. It doesn't really add another verb (to be) to the sentence. Not in Polish, definitely. Present Simple and Present Continous are translated the same way in 99% of the cases. Your sentence sounds kinda like "... although I am drink wine" It just doesn't make sense, because you added an additional verb to it.