As you can see there is quite a bit of degradation of battery when viewing original Design Capacity: 2850 mAh vs current Full Charge Capacity: 1917 mAh-which would mean much shorter battery useage. That 'could' account for the laptop not really reading the remaining capacity correctly and shutting down.
I would also test it with power cord plugged in-just for comparison-nothing really important. Also test power adapter, system board to see if any anomalies.