AT&T Wireless Cell phone not a year old an ATT refuses to replace it Chattanooga, Tennessee

If you bought the phone less than a year ago, it will still be covered under warranty.  However, the warranty will not cover physical damage above and beyond normal wear and tear, nor will it cover 'liquid damage' that comes from the phone's sensitive electronics being exposed to water/soda/beer/etc.  Those kinds of damages would be covered under the wireless phone insurance that the agent tried to upsell you when you got your phone.

It sounds like your phone has some physical damage to begin with, and the description of the phone turning off all the time sounds like it could be a number of issues, possibly liquid damage.  If they are refusing to cover it under warranty (if you called 1-800-801-1101, that's the warranty number), then one of the questions that they asked you led them to believe the same thing.

Warranties are against manufacturer's defects, after all, and what ou are describing sounds well beyond that.

You do have a few options from here:

1.) Send in the old phone, and hope that they decide that it's warrantable, risking 150 bucks in the process.
 
2.) Buy a new phone at the no-commitment price (read: expensive)

3.) Ask about an 'Exception Upgrade' at your local AT&T store.  This will allow you to sign a new two year contract before your old contract has expired, allow you to get a new phone at the new-contract price, but with an 'exception fee' which is $75 for most phones, $200 for iphones.

4.) Go to a local store and ask to see their go-phones.  These are low-cost phones (some as low as $30) that are generally set up for pre-paid customers.  These phones can be bought outright by customers that are already in contract without signing a new contract, and can accept the SIM card from your failing phone, getting you a working phone for minimal cost.

Personally, I would go into the local store and try the gophone (#4) option...and while you are at it, have the agents take a look at your old phone, just to see what may be really wrong with it. It may indeed be able to count under warranty, but the warranty department will listen to them a bit more, since they are holding the phone in their hand.

Good luck!