No, it only charges while on. I don't have quite the problem you have blimpsgo180. I'll check with dmesg when I get the chance.
I took the batter out the other day though to see if it runs without the battery in it and i think the battery might be puffing up, not sure, I'll take some photos when, once again, get around to it. I have always had trouble closing the battery door, it always seemed tight, but it seems tighter now. Not sure if it is in my head though. I might get a new battery while there is still stock. I might ask ED if he thinks they will still be in stock when the Pyra ships, maybe I'll get both at the same time.
I'm actually impressed it still works though, I think every phone that is this old had the battery go out and not hold a charge anymore after just 5 years. It would be great if something is just wrong with my battery though and all I need is a new one.