iTunes says I've run out of apple ID authorizations, but on apple ID website it says I only have two devices registered.

What website are you going to ? Computer authorisations and associated devices are two different things, authorising only applies to computer's iTunes (and that's the only place for authorising and/or deauthorising and seeing the number of authorised computers), iOS devices aren't authorised to an account.

https://www.icloud.com

This problem is happening when you try to transfer a purchased album from iTunes library to a device -- but the purchase definitely has not been used on 5 devices yet

Authorisations only applies to computers, not to iOS devices. Is the account that bought the album authorised on that computer, does the album play on the computer ?