They are grammatically OK but as you suggest, they sound awkward. People might talk about "letting themselves do" something when they are recovering from a difficult experience and learning to trust in their own abilities again. "Study" is not the kind of activity I can see any problem with. e.g. After I broke my leg, I only let myself walk without crutches after several months. The second sounds better like this : "Shall I try studying now?" It still sounds like a funny question as study is something you normally decide to do or not, not something you "try" to do, or ask permission to do. Though you could say this: "I tried to study for 1 hour but I was too distracted."