I’m Melissa, a Shopify Guru. For the 'Free Shipping' rate, you would like to add that can be set up manually within your shipping settings if you would like to. When setting up your manual shipping within the Shopify admin, there are two conditions in which a rate can be created; Price-based http://bit.ly/2qykIw7) or weight-based (http://bit.ly/2qycswd). Those video links will show you how both methods can be created.
If you need to set up a more custom shipping method for your product or collections, there is a range of apps that you can take a look and see if they would be a good fit for what you would like to have: http://bit.ly/2qyjEbQ
To use some of these apps you may need to have Carrier calculated shipping enabled on your store in order to use the functionality needed for the app. That may be why the Better Shipping app was not working correctly, in order to avail of the apps full functionality you would need to have Carrier Calculated Shipping added to your plan.
If this is something you would like to add to your plan, it may be possible. It would be best to reach out to our support channel if you want to add that and they will be able to help you get that enabled. Hope this information helps! You can reach out to the Guru team https://help.shopify.com/questions if you have any questions.
Have a great day!
Melissa | Shopify Guru