When setting up your service accounts you want to follow the principle of least privilege. Basically this means that you only want to grant the necessary rights to your service accounts to do it's job and nothing more. I have never come across a reason that I needed my service account to be a domain admin or a local admin. I simply grant the necessary permissions to the account and that is all.

The reason for this is security, plain and simply. If you service account gets hacked you want to limit your potential damage by limiting the hackers surface area. Domain admin and/or local admin is a pretty big surface area.

Now a couple of people have asked if the new account has read/write access to the location of the Tlog backups. Does it? Start there. Also, any changes you make to your service account needs to be done through SQL Server Configuration Manager not by going directly to the service itself.