I have had several Casio EX-S600 cameras (eight to be exact, they can be had relatively cheap on ebay) that all eventually suffered from an issue similar to what you describe, rendering the camera unusable. From the research I've done and the camera operation I have witnessed myself, I can say with almost 100% certainty the shutter is sticking. You describe distance photos as being washed out (white) and close up photos as OK. If your shutter is sticking, distance shots allow much more light into the aperture than the close up shots, hence the overexposed/washed out/"white" photos. However, as the shutter operation gets worse, and it will, even the close up shots will come out overexposed because the shutter will eventually hang wide open all the time. Since parts are mostly readily available on ebay, I attempted to make a repair myself (after I called a camera shop and found out the repair would cost many times more than the camera was actually worth). I have repaired several small electronics myself; iPhone batteries, iPod screen, but this camera... YIKES! Very densely packed internal components, to say the least. I'm still waiting for a step-by-step for an aperture/shutter replacement. Unfortunately, who uses actual cameras anymore, right? And then there's the throw away nature for these types of electronics that hinders the DIY segment... sigh...
I also wanted to ask, do any of the photos you have recently taken with this camera since the issue developed look weirdly pixelated, or have strange lines running through them when you view the shot on the display?