Att offers insurance but you have 30 days from purchase of the phone to add the insurance.
Without insurance you have the 1 yr manufacturer warranty so if you call att they will swap it out but with a refurb.
I'm assuming the battery is charged and holding a charge? Do you know what the battery % is at the times the phone is shutting off? Do you notice it happening more when you are doing something specific with the phone or just whenever it feels like it? If you don't want to do a refurb swap you could try a factory reset and see if that helps.