Battery Won't Charge, stuck at low battery or constant reboot [3GS]

[Infantry53]

Received this iPhone as a hand-me-down, was told it would not hold a charge or restore. When I hooked it up to the computer the phone would get very hot after a short amount of time and was stuck at the restore screen, after every restore would get an "Error 29" which I read was due to an incompatible battery, the phone would however be able to stay at this screen unplugged for a short while. I ordered a new battery and the phone would not even get past the low battery indicator, and does not get hot anymore at all, I put it in DFU mode and as able to complete the restore without any errors, phone would still not charge, I put the old battery back in and got it to the activation screen but will not charge. Now with the old battery when plugged into the wall it restarts every 5 mins, plugged into the computer, the apple logo comes on and off every 3 seconds turning on and off. With the new battery the phone will not get past the low battery screen. Apple said they would swap for 79.99. Is it worth it, or should I try something else?