replaced the battery, not helping

the battery was not holding a charge, so I got a new battery, which seemed to help for about a week. Now it will intermittently power on and is flashing a grey dead battery logo even though it is plugged in to the charger. Then sometimes it will power up and say 100% battery charge.

I tried to do a reboot- most of the time I could not get it to go to that menu, but once it did and so I chose reboot. It didn't help.

Any advice?

I took it to the ATT store and they suggested I buy a new phone? Is that really my best option?