Is there a file sharing service that allows me to create admin-level logins for a handful of users and allow limited access to a wider group of users?

HyperOffice is a great option. Give everyone "admin level access" or "subordinate admin access" or "simple user" access, or even create custom profiles to apply to users. Plus you get robust file storage, sharing and collaboration features. http://www.hyperoffice.com/online-document-management/

Do have a look at http://www.oogwave.com A very good alternative to Google Drive, helps you manage all files in collaborative folders and sub-folders inside workspace, with user based permission on folders. Unlike GDrive, all folders are by default shared to all members in a workgroup, you have to restrict them for users if required.