This definitely sounds like a battery would be a good start, but we can check for a few clues before actually repairing if the phone will stay up for long enough to do so.
If the phone will actually boot (I couldn't tell from your description if it ever booted properly and stayed up) I would check two things: your battery health, and if there are any panic logs present. Battery health can be found in the battery section of settings. And kernel panic logs can be found in the analytics settings. You can use the [[iPhone Kernel Panics|iPhone Kernel Panics|new_window=true]] page to locate them and there is some info in interpreting them on the page as well, but if it comes to that, it might be better to post them here so we can have a look.