Jessica, except for Athens, towels are widely available, and often very cheaply ... in Nafplio, venture a block or two outside of Old Town, and they'll be dirt cheap ... 4 - 5€. I remember a friend & I at St. George beach adjacent to Naxos Town, neither one of us had brought a beach towel, we vied for title of Ugliest Beach Towel @ €4. My friend won but I was a close runner-up. However, in other places I have bought 2-3 other towels (one is blue/white Greek flag, another says "Ellas" in Greek letters, another is all dolphins) which I proudly sport at my local health-club pool and elsewhere, to admiring comment.
BTW, you say you're doing carry-on (I assume a rollaboard 20" or 22" suitcase), you also can bring on a "personal bag" the size of a sizable "tote" ... For my Greece trips I bring a tote with shoulder length handles, about 14" x 20". Inside, I carry my "dagbag, phone, magazine, paperback, snacks, AND my lightest-weight beachtowel, which then becomes my neckpillow for sleeping....