Well if you do a full restore in iTunes you MUST NOT restore from any backup of any kind to fully confirm that it a a hardware problem. You will have to restore and setup as new device don't restore from iTunes or iCloud backup.
I don't know what carrier you have but I have Verizon and when I got my iPhone replaced damn near 15 times they sent me a phone in the mail that came in one day. After I received the new iPhone I had to have the old one returned within 5 days, and the box came with a sticker and was free to mail. Perhaps your carrier can do the same.
You can also go through Apple and do kind of the same thing, but they charge you debit card for $600 (whatever the full price is) and when the old phone is sent back the charge is removed. Called Express lane I think. But 100% on that.
I always just went through Verizon, they sent me a new phone the next day, in which case I used the same box to return the old all completely free with no payments of any kind.