Hi Nate, thank you for your question!
Deploying the Staged version of your website will correctly update the links to the pages in your main menu, but you will likely need to update any link information that was set in your page content through the editor.
In the future I recommend trying out our Cloud WordPress system through BoldGrid Central instead of the Staging plugin to handle staging for your website. Using Cloud WordPress will allow you to test changes in an environment that is separate from your main website so you can try out new plugins or themes to test compatibility without worrying about affecting your main website.
Please let us know if you have any additional questions, we are always happy to help!