Battery Sometimes Working - Bad Logic Board?

First I would download Coconut Battery (it's free, just Google it) and take a look at what it has to say about both your batteries. You want to know for sure that they are not the problem, and this program will tell you the remaining capacity, load cycles, etc. All batteries go bad eventually, and most battery issues are caused by bad batteries.

I would borrow a friend's AC adapter and see whether or not the problem still occurs, preferably while you are using a known-good battery that you have tested with Coconut Battery. It could be your AC.

Next, I would suspect the battery connector. These sometimes come a little loose and act strangely when they do, so it may be worth re-seating it. Your description of squeezing the computer near the battery may be an indication that this is the problem. Battery connectors have long pins and they need to be pushed further down into the socket than may initially seems necessary.

You can also replace the battery connector, which is a worthwhile step if it comes to it, because it's only a $10 part on eBay. Just make sure to buy one from a reputable vendor, because most sellers don't test these, even though they claim tested and working. Also, there are two types, depending on your logic board -- Energy Star (gap between the pins) and non-Energy Star (no gap), so make sure to get the right one.

Good luck!