S4 won't connect with usb AND broken screen. Need files from phone

My S4 will not connect to my computer when plugged in using the correct samsung cord AND the screen is shattered. How can I load the files from the phone? Remove the hard drive from the phone maybe and put it in another without ruining the new replacement S4? And if there is no possible way a all to retrieve the files then what can I do to make sure no one else can retrieve them after I send the phone back to the insurance company? I have 4 days to get a solution. Thanks!