I already paid for the app but I can't access my favorites

It is really so. After the recent update you need to have Ultimate-guitar account for using your favorites (if you don't have Ultimate-guitar account). 
But don't be afraid, it is free and simply to create it. Just go to account section of the app and press "sign up".  And you will avoid losing your favorites. 

If you've just forgotten your Ultimate-guitar account info, just read this article: I forgot my password

Note 1: your Apple ID password is not the same as your email password. If you don't remember the password for your Apple ID, please use "Forgot Password" option in your Apple ID account.

Note 2: your Apple ID is not the same as Ultimate Guitar account, please don't confuse them.  

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