In order to use the backups, you'd need to do a hard reset (which wipes everything on your phone), then log in to the same MS account and restore from the backup. If you have multiple MS accounts and one of them is working on your phone, you can copy the contacts over. To do so, follow these steps:
Log in at https://outlook.live.com/owa/?path=/people with ...
I'm not sure if this will work with your account scenario, but I have done a similar in the past with some accounts of my own.
If you already have an old and new microsoft account, migrate the xbox gamer tag to the new account.
Log in with the account you want to change
Go to Settings and select Account
Scroll right to Your Information and select Account ...