Welcome to AF!
I presumed from the video that you have the Verizon SGS4, so I've moved your post accordingly. If I'm incorrect, let me know and I'll move again.
I was in a similar situation with one of my handsets earlier in the year. I uploaded the TB backups to dropbox, then when my replacement phone arrived, I rerooted, restored and it worked with no problems. I imagine it will be similar for your SGS4, but hopefully the guys here will be able to assist.