I went into iCloud in Settings and I noticed that my phone still said "Restore In Progress," though as far as I could tell it was back to normal (minus the annoying apple id prompts). I clicked "stop restore." It warned me that I would lose whatever had not been restored yet. I decided to take the risk.
Then from within the same menu I told it to backup my phone in its current state to iCloud.
Once the backup completed... voila! No more prompts. I don't know what happened and I don't know how to explain it. Maybe the backup has kinda come full circle or something. It restored from the cloud, then backed up again and now everything is legit...? I don't know.