Unless there's another symptom such as failure of the system to recognize the AC adapter or failure to run on AC power (be sure you check the adapter status in setup -- F2 at powerup), the battery is almost certainly the issue.  
Power light is flashing yellow and system turns off immediately (i.e. no formal shutdown) if power cord is disconnected. In BIOS->About Battery, it says "Health: communication error". However, I ran the full diagnostic suite from the Dell support site and it found nothing wrong. Additionally, Windows still says the battery is still good. I'm not averse to replacing the battery if it's actually faulty, but how can I determine if the problem is the battery or the motherboard (or possibly even something in between)?
Latitude 7370 with battery issue