you can’t make the @ record a CNAME without deleting all other resource records for @, and you can’t do that since some (like the NS records) are required for proper DNS functionality.
You will need a host to perform the HTTP redirection from example.com to http://www.example.com for you, to which you will point A (and AAAA) record for @.
If your DNS is hosted with GoDaddy, then they have a free service that will do this for you. In your GoDaddy domain manager, look on the left hand side for “Forwarding” and click “Manage”. Then set it to forward example.com to http://www.example.com and update your DNS to support the change. You should leave the Advanced Options at their defaults.