Short battery life

Since you have a new battery and charger then we can eliminate that part of the hardware. That leaves an internal camera problem or how you are using the camera.

The #1 power drain is the flash. If you are taking a lot of pictures with the flash it will quickly drain a battery even if everything is working perfectly. You can try turning the flash off or down to see if this helps. The other thing is using the camera to look at your pictures. Not anywhere as big a drain as the flash but it does use some. As well using the camera to take videos or view them is higher drain. And taking big high quality pictures is more drain than smaller ones.

You may have an internal hardware problem in the camera itself. If the flash doesn't make a difference then I'd be inclined to think its the camera. If it looks that way to you after seeing what it does without the flash and so on then I would first check to make sure you have warranty and send it in to your local repair depot. You can find that by going to the Kodak web page and looking under support. You may want to talk to them and see if they can suggest anything else. And make sure you know what the replacement cost is before sending it in. Usually the repair cost is fixed but by the time everything is paid is shipping then it may be cheaper just to go get a new camera.