After a trip to my local Apple store, my Genius Bar helper Sugar Bear informed me I had a dead battery...
Sugar Bear told me about iFixit, and within a couple days, everything I needed arrived on my doorstep. 15 minutes later... computer that runs like new. No sweat, no problem.
Definitely keep your screws organized. I just lay them out in the box the replacement battery came in as I removed them. Takes all the guesswork out of how to reassemble the laptop. Good luck!