If you are completely blocked from signing in and unable to change your sign-in info from the aka.ms/mfasetup page, your options are to either have another global admin reset your info to let you in, OR if you are the only global admin for the organization, you will need to contact the Azure Data Protection team to get help accessing your tenant. Their number: (866-807-5850)
The behavior you described generally happens when you have two separate accounts (one personal and one work) with the same email address. You can set up your personal account to use a different email address if you don't want to get it mixed up with your work account. (Ref: Which account do you want to use?)
Converting or merging Microsoft accounts is not possible to do, but you do have the option to create a new work account.
https://support.microsoft.com/en-us/account-billing/linking-microsoft-accounts-4044c1bc-f436-60eb-9f6d-2bf64a7cc47f