To solve this, I ensured all my projects used the same version by running the following command and checking the results:
update-package Newtonsoft.Json -reinstall
And, lastly I removed the following from my web.config:
<dependentAssembly>
<assemblyIdentity name="Newtonsoft.Json" publicKeyToken="30ad4fe6b2a6aeed" culture="neutral" />
<bindingRedirect oldVersion="0.0.0.0-6.0.0.0" newVersion="6.0.0.0" />
</dependentAssembly>
If you want to ensure all your Newtonsoft.Json packages are the same version, you can specify the version like so:
update-package Newtonsoft.Json -version 6.0.0 -reinstall
Note: To open the PMC in Visual Studio, click Tools -> Nuget Package Manager -> Package Manager Console. Thanks @Rose!