The solution here is to create a new copy of the Action. You do this by cloning the Action.
From the quick edit module of the Action in your new App, tap "clone" under the Action Settings icon and this will create a new Action. It will be identical to the previous Action except that the word "Copy" will be added to the name. Give it a better name and you are good to go.
Just delete the original Action from the new app. It will remain in the Library and in any other apps that use it.
Note: Having a different name is what makes a clone different from the original Action, though once separated the two can be changed independently.