OK, well,sorry about the late answer, but I was working off town (as I usually do, in fact that was one of the reasons because I got my Z...)
well, about your issue... the HOT sympton info could be some bad or could be worst.... ( I doubt the last because you said that the only thing before the troubles was leave unplugged your z, so I think more in a fault battery...)
well, I would suggest others steps in case that the previous 1,2 or 3 didnt help...
the issue of HOT may suggest a Battery with its internal short circuit, (or a short circuit on some part of the charger circuit of your Z motherboard... ), well, because that is a MUST to determine WHAT is the part that made the hot so ,hera are some few tips that could help you...
4) Try to find someone near to you that had a Z same as yours and try a test with a normal operating battery (this could be not easy because the Zs are no too much spreaded.... hummmm a theme to thing.... ) well if you didnt have a good Z near to you try the following...
5) Separate your Z and you battery, lets then for 24 hours and mount them together... BUT FIRST lets a light cold of both parts, some tips on this.. put the Z and the battery in separate sealing type plastic bags (like ZIPLOCK bags....), inside the bag put some silica gel bags (the small white bags that came inside the bags with electronic stuff for absorb the moisture on the packages during delivery...), THIS IS VERY IMPORTANT!!!!... put a silica gel bag in the battery bag too.... seal them and put on the refrigerator around 20 minutes to cold down both.... check than them are cold at the touch of the finger,then CLEAN VERY WELL any moistrue of the Z and the battery with toilet paper or similar then setup the Z with the batterry and connect the adapter cable to the Z for few minutes and at the first signal of hot, unplug the Z and hurry check what part ( the battery or the Z are more hot than the other, ) so if the battery is hottest that the Z may be you had some luck and only your battery die, but if your Z is more hot than the battery then may be you had to service your Z... or replace it...
5) for be sure , at last this is a advice that I read somewhere (i Use it to diagnose the battery of my notebook and the battery was dead)... repeat the procedure of colding but ONLY FOR THE BATTERRY, and put some silica bags on the same bag together with the battery inside the same sealed plastic bag and put in the refrigerator for 24 hours, then CLEAN VERY WELL any moistrue of the battery surface with toilet paper or similar...and retest the whole thing again, if for some minutes the things work and after the things goes hot the almost for sure you had a die battery...
6) About battery replace, I didnt know about similar, but by know some cellular phones use batteries that are very similars to the Z\'s in a very suspected way...so If you need to change your battery, I suggest you that start by searching a replacement in a large cellular pnoe battery store and may be you could find a nice replace... the phone that I saw with similar batteries are motorola, but never tested of know somthing more, I noted it some weeks ago when I had some trouble with my cell pone and had to go toa cellstore for some service, and the girl borrow me a motorola 130 or something like that but in that time I didnt had my Z with me or rememnber the voltage of the Z...
GOOD LUCK and best regards!
Xaser