Along with the sign-out/sync steps check free storage - older Kindles need extra headroom to unpack large sets and and a nearly full device will hang. If indexing is stuck on a bad file, delete that title, restart, wait a bit for indexing to finish, then try downloading a single volume first. If that succeeds, either stick to individual books or use the Download & transfer via USB option in your account so it sends a format tailored to your exact device.
