Batteries dead after very few shots
It has nothing to do with your camera. At least not when your camera is not becoming hot. Some batteries can work for thousands of hours, and exceed the 1000 recharge cycles most battery firms are talking of. But the don't give any guaranty a battery will last that long.
Some camera manufactures give one year of warranty on the camera but only 90 days for the battery. So it could be your batteries are old and worn out already. You can use 2 AA size NiMH rechargeable batteries, to check if the do a better job. They are relative cheap, and should last as long as the original battery pack that came with the camera. Please, even when you are not using the camera, you should charge the batteries at least every 2 or 3 months. Then they stay in a better shape. Never use NiCD rechargeable or manganese (Zinc-Carbon) batteries.
Even the very expensive alkaline batteries won't last half the time a NiMH could work. GP batteries, makes ReCyko+ rechargeable batteries, that not exact do have the capacity of the NiMH batteries, but they can hold the charge much longer. You only have to charge them once a year, when you did not use them. and still after a year they should have enough power to make a bunch of pictures.