Do I need a monthly Apple Music Subscription AND an annual one?

You shouldn't.

Check your subscriptions.

Purchases - View, change, or cancel your subscriptions.        

Purchases - Subscriptions not billed through Apple     

Normally, all sales are considered final, but you can ask for a refund.

If you have received a confirmation email, click the Report a problem link at the bottom.

Purchases problem.      

How long refunds from the App Store or other Apple services take.   

No.

If you have one Apple ID and access music on that same Apple ID, you shouldn't need more than one subscription for the same thing.