I had the same problem with my 35 months old iPhone 5. I took it to the authorized service center.
I told them that I suspect it to be the swollen battery issue.
After a diagnosis of few minutes, they agreed and told me that it will take a couple of days for the phone to get replaced. The techie also told me that this was the last month for me to get the phone replaced as the coverage was for 36 months.
I got a brand new iPhone 5 yesterday. I had to survive with a makeshift phone for 2 days though.