Most laptop batteries don't last long! thats why even on the extended laptop warranties they don't cover over 12months on battery.
The trick to help batteries last longer is not to leave them plugged in constantly in the laptop - let the laptop completely discharge and recharge at least once every week or two.
as for a solution - you can usually buy a new original on ebay for fairly cheap. I don't believe its an overheating issue - if it shuts down due to heat it would shut down even while plugged in