Does Shadow work when my internet connection is turned off?

No, you will not be able to access your Shadow if you lose internet access. For a fiber connection, this problem is unlikely - fiber failure rate is extremely low (it does not exceed 2% for a connection older than 1 month). This may represent a barrier for some but in all honesty, when you do not have access to the internet, do you actually use your computer? Or you are the type of person that is going to do something else seeing as your odds of being able to log into your online game are gone? Or maybe you're going to spend all your energy in making sure your ISP is re-establishing your connection?