How long does this battery last on a full charge? If it lasts just half as long or less from that of which when it was new, I'd replace the battery.
I have seen laptops with batteries that have aged and are on their way out cause a similar issue. The issue is caused when the battery is unable to keep up with the power demands of the laptop and the charger is unable to keep up with the power demands of charging the battery. As the battery drains and the charger is unable to charge the battery at the same time of powering the laptop the laptop will eventually crash or give low battery warning. The charger at this point is burning Hot because it was trying to keep up with the load and couldnt.
If you leave the system off for a period of time battery will recharge and the system will behave the same the next time you use it where the AC supply cant keep up with the demands.
The reason for not being able to keep up with demands is usually because the battery itself has a cell that is bad and is fighting the power supply to take a charge. This usually makes the battery hot when it is charging, although I have seen some that have had a bad cell only be mildly warm during a charge in which it would seem normal for a battery that is charging for feel warm.
Do you have a spare battery for this laptop to swap and see if the problem goes away?
Some laptop batteries have a LED indicator on them that you can use to test the battery outside of the laptop, although many dont. If this battery has this feature I'd push the button on the battery out of the laptop and see if it shows a good status for the battery cells.