Hi Kevin,
Okay, there are a couple of different possibilities here. This would seem to be related to the amount of power being drawn by the computer; when you run a video game the power consumption goes way up with the heavy use of graphics.
Now, most computers can run directly off the AC adapter even with the battery unplugged. Not all of them, however, and I can't tell you if yours can or not. With the ones that can't, power always comes from the battery even when it's being charged. I mention this because if it can only run off battery power and you've got a battery that's nearing the end of its life, it won't be able to hold enough charge to supply all the power needed to run the graphics and charge the battery.
The other possibility I see is that your AC adapter is getting old and can no longer supply all the current needed to both charge the battery and power the computer under load.
Either way, it sounds like a power issue; either the battery or the AC adapter. The most straight-forward thing to try is to buy or borrow another AC adapter and see if the problem is still there. If so, then I'd be seriously considering replacing the battery.
Let us know how it goes.