And just to be clear you’re using the correct user levels for this — the default permissions in WordPress allow users with contributor-level access to create posts and submit them for review. Then, an editor-level or administrator-level user account can see them in “pending” status, review, edit and publish them.
With the plugin above, and using contributor-level user accounts, your WordPress site should provide the functionality you’re looking for.
Here’s a quick walkthrough video I created outlining the ‘contributor’ user post workflow: