These comments are being pushed to the emails of some of my followers--as if new content has been published.
How can that happen via your blog, when you moderate comments and approve every comment posted to your blog?
You don't have to require a log-in or an email address from commenters, but if you don't then you have flung doors to your blog wide open to spammers. See more on that at the end of my comment please.
This page > Settings > Discussion is where you set up defaults for comments.
Other comment settings
_Comment author must fill out name and e-mail
_Users must be registered and logged in to comment
Note that the prompt to submit and email address and a website URL will still appear but providing an email address and logging in are not required.
You can post instructions for your visitors in a text widget if you wish to set up anonymous commenting. That's easy to do. You copy and paste this into a text widget:
Readers: Note that the email address and logging in prompts do appear for comments on this blog, but providing an email address and logging in are not required.
P.S. Do be sure to set up full comment moderation ie. "An administrator must always approve the comment" because anonymous commenting is bound to increase the amount of spam your blog will receive.