iPhone refuses to sign into iCloud

A few days ago, I noticed my iCloud data wasn't syncing properly, so I decided to sign out. This took much longer than it should have, so I cut it short with a hard reset (hold power + home). Later, when I tried to sign into iCloud, it took a long time as well (30 mins?), so I cut it short again with a hard reset.

Now my iPhone seems to be in limbo: trying to sign into iCloud leads to the normal "Verifying..." screen, but stops without signing me in, leaving my username and password in the input fields. As well, all my iCloud data is gone (Notes, Contacts...), but there are still empty options for them in their respective apps (e.g. the Notes app has an iCloud portion with 0 notes).

I've verified that my Apple ID isn't locked through appleid.apple.com, as well as icloud.com.

Any help?

iPhone 5S, ME341LL/A, iOS 10.2.

EDIT: I'd like to avoid restoring if possible: my cable is pretty wonky and I'm worried about the restore being interrupted somehow.