Hi sarahblisss,
Welcome to the Apple Support Communities!
I understand that you have downloaded an album from iTunes, but when you try to play it you receive an alert that you must be signed into Apple Music to access it. From your original question, it sounds like you are already signed into Apple Music and are still unable to listen to that album.
To begin troubleshooting, I would suggest disabling Apple Music and iCloud Music Library. After disabling, sign back into the services as described in the attached article below.
I am uncertain if this issue is happening on an iOS device or iTunes on your computer, but the following article does explain how to sign in and set up Apple Music and iCloud Music Library on either an iOS device or computer. Please test playing that album after you are signed back in and your library is active.
Check your Apple Music library settings - Apple Support
Best regards