I see an "edit" link and a "delete" link next to all of my shipping options.
When you say:
If I try and delete the existing profile it says I can't because there's product attached to it. This means I have no way of adjusting the cost of shipping.
It sounds like the system can't delete any shipping option that is in use by a listing. You would first have to edit any product that is using that option until no products use that option, and then you should be able to delete the option.
That's my best guess on what's going on.