Good Grief! 20 Euros is one taxi ride in NY... don't let it force you into an itinerary you don't want. I would vote for FOUR nights in NAxos, and 2 in Folegandros. Ive been to Greece 13 times (yes, I know , but Im not rich, I just go nowhere else) -- have visited more than 25+ islands, a handful of them 6 or more times... and Naxos 14 times. So far it's escaped being spoiled, (because its airport wont take huge jumbo jets, only small turboprops, and its harbor can't accommodate cruise ships. YAY).
Folegandros is all about its high village, chora, which is perhaps THE most perfect Cycladic village ever... and Ive seen scores of them. I walked around for 2 hours one day, giving myself the challenge of finding a single corner or angle that was unlovely. Not one. Stayed at https://www.anemousahotel.gr/en/article.php?cat=2 -- in May, V moderate, a dream infinity pool, felt as if I were in a perfume ad. I was visiting friends of mine who that year were teaching in the Gymnasium (Hi School) there... they showed me around. Not so much besides Chora (and the climb up to the church on the peak ) ... but it's worth 2 night stay.
Also spent 3 nights in Syros one year... wanted to see its famous Ermopoulis Square which is indeed impressive. However, I was not "taken" by the island... others here strongly disagree with me. It's 99% Greek, it's their island, and I felt totally excluded .. no sociable interchange even with waiters or baristas at the beach. SOO different from Naxos, which is friendly wherever you go ... not just in port town or beach... but up in hill villages ... one time I sat on a fence for half an hour & watched an old geezer shear a sheep, and we talked (he talked greek, I talked sign language). Others may have gone in a group, provided their own sociability. It was the first & only Greek isle I haven't experienced welcome... Greece is the friendliest country I know.