Yes, this is correct. "I finished the tea you sent me..." THE because it was one particular type of (blend of) tea. And yes, they (those blends/varieties) are available in the supermarket. I was in Russia 8 months ago, for the first time. They had Lipton tea and all the familiar English brands, but after the name "Lipton" the rest of the package was in Russian. I like your name. It reminds me of the song, Coin Laundry. No inferences intended, I just like your name. Phill

Since you are talking about a variety of teas, the sentence would be more clear if you used "teas" instead of "tea." I hope this helps!