Ok, so update: (I'll try to keep this short!)
The pipes were fine, no leaking, no leaks where any of them join, etc. From what I can understand, the water wasn't completely going into the drain from the overflow and it was causing the water to get between the tub and the wall, therefore, leaking onto the ceiling below. Like I said, I don't completely understand all of this but I get the point. I ordered a gasket from the place someone suggested - Ultimate Overflow
) and even put it in myself (whoo hoo!). I filled up the tub and recreated the scene of the "crime" - before fixing the drywall of course (the drywallers will be here Thursday!) and nothing happened.
I feel confident in the drywall being installed this week now that the problem seems to have been fixed! I just hope I don't have to do that ever again... Thank goodness there was nothing in the room below or that would have been even worse! Now I just have to go to Home Depot for carpet...
Thank you all for your support!