Alright, reporting back in on this one...
So, System Profiler says my battery is in Normal condition with 689 charge cycles, says the battery is installed but that it is not charging. So there's my problem. I reset the SMC and nothing really happened. Same situation, powers computer but does not charge battery. I'm pretty sure I reset it properly, so I don't think that's the issue.
I tried running the Apple Hardware Test but it kept booting normally, so then I dug up my old Snow Leopard install disc and tried again, but it still just booted normally? Not sure what I'm doing wrong here either.
On the final note, I can certainly replace the board, I just would like to do any troubleshooting things before I start chucking money at it. I'm not sure if maybe just a wire is loose or something simple, or if the thing really is blown. I've had this problem several times in the past few days. I couldn't even get it started at first, so I switche the battery out of the other MacBook and then this one worked but wouldn't charge, so I plugged the charger in first and then stuck the battery in and it worked great? This whole situation is just confusing and very iffy...
And finally I just wanted to thank you for all your help, it's great to see a helpful and friendly Apple Support community!