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The camera might have gotten hot, but are you sure that it was not the battery itself that was getting hot? so I ask, how old was the battery in the camera, when the overheating and other symptoms occurred, please?
So the battery was old too?
Then try replacing the battery.
Rechargeable batteries do get weaker with time, and eventually fail to deliver enough power to start up a device like a digital camera; they then need replacing. The replacement battery needs to be charged up for 6+ hours before its first use, owing to its having been on a shelf for a considerable time.
(Of course, check for the first few minutes of charging up, if the new battery charges within the camera, to be sure it isn't overheating also!)
(If the battery charges outside of the camera, there is no worry of that sort involved, of course.)
OK. Let's see what happens. (Thanks for letting me know.)
Try removing the batteries, and leaving the camera on a (dry, non-humid, non-dusty) shelf for overnight. Then put the batteries back in and try the camera again.
Then try cleaning the terminals in the battery compartment, after removing the batteries, first by rubbing off with a fresh pencil eraser, then by wiping the terminals off with cotton swabs moistened in rubbing alcohol, then letting them dry off for 5+ minutes with the battery compartment door open, then try turning the camera On again.
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