All you need to do to solve this is to rewrite the shipping render total. That is done pretty easily and lets you make changes whenever needed.
You may use your current customization module, or create a new one from scratch - here you will need the rendering block (instead of the default one):
https://gyazo.com/f1dd6faaee9cb5540432cf88710f024a
This is the block content:
https://www.gyazo.com/220b5c53f897dcb4558b1177af70dcb5
Here we change only the template, so you can inherit it from the default one and rewrite only the parts you need. Alternatively you can create a new one. It's up to you to decide and depends on your requirements for the block.
Next, you will need the template mentioned in this block:
https://www.gyazo.com/260227ea9345a0f23438b0ec60c3e6ed
You can get it by copying the default (app/design/frontend/base/default/template/checkout/total/default.phtml for the basic theme).
Right in the template you only need to hide the price (using the Hidden tag) or remove it.
https://www.gyazo.com/9b3b61fc3e9793d8d1de1d1b58589855
And finally you only need to modify the layout of the objects you need to use for the directives. For the default shopping cart and checkout it will look like this:
https://www.gyazo.com/df2a27378d89e0d5b111ab7d7772051b
And this is what you should get: https:www.//gyazo.com/b3208c432a574a35303738e4119ffab5
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