The Shrink Wrap modifier does not work very well with separate mesh objects.
There are two options.
You can use the Boolean modifier to union the ax head and the handle. To make this work the handle must be wider. The axe must be solid. Your model has a gap between the head and the handle. This causes problems.
Alternatively, you can duplicate the axe and create a remeshed dummy axe for the Shrink Wrap modifier. To make this work you need to close the hole in the ax head to prevent inner faces and a mess. When the hole is closed you can remesh the object with the Remesh modifier. Add and apply it.
To wrap the ribbons scale them down and place them inside of the head. Then select
Project for the Wrap Method of the Shrink Wrap modifier and give it a negative Offset like
-0.02. The Target should be the dummy object (option 2) or the solid axe with the Boolean modifier (option 1).
Location for the ribbons (the Shrink Wrap modifier is disabled):
Result with the remeshed dummy axe (option 2):