Trying to add a free shipping option

I have created a free shipping option in my store that kicks in for order above a certain amount but I haven't figured out a way to have it become the default once the amount has been reached.
What is currently happening is that the little cart window showing the current amount and shipping shows the price based on my weight band method even when the free shipping amount has been reached.

I am not sure if there is a clever way with current tools to do what I want but if not would it be possible to show the lowest amount of any shipping method that matches ?

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