Don't game without ac connected. For me battery for these types of laptop is just like a UPS, to give you time to save your work and do a proper shutdown before power runs out during a blackout.

Is there a way to test my latop battery to make sure it is not faulty with like a voltmeter? Is there a way to check to see if there is something wrong with my laptop hardware that will not allow my laptop to draw enough power from the battery? Because the battery keeps its charge and I am able to use it in class for MS Excel and using a website asks questions during lecture. No problem!
I get a problem MOST of the time with Runescape, which is not very demanding. I can get to the log in screen and chat with friends in the lobby. If I start the game, it shuts off.
If I start other more demanding games like Starcraft 2 or Crysis2, I don't even get to see the initial loading screen or game manufacture screen. It just shuts off.

I would like to avoid having to send this battery back. But if that is the only option then I suppose I will this weekend. If I get a 3rd battery and this happens again, what should I do?

I would get another battery first if it's not too expensive, and see if the problem goes away. What does your battery indicator say? 97% available. plugged in, not charging?

Use something like HWMonitor that can tell you how many MWH your battery has left. Hopefully you didn't get a knock off that only has something like a 5,000 MWH lifespan.