I have never even heard of anyone having a problem with this.
Everyone else is just saving games and exporting characters, then using those files when they want. You really haven't given any information as to what's happening.
Inside the zipped folder you downloaded and unzipped (in the exe version), there should be two txt files (readme and disclaimer), one folder (jre) and the exe file itself. Upon saving a game or exporting a character, a second folder called "data" will be created that contains the error log (if the size of this file baloons, delete it), an xml and two sub-folders labelled "characters" and "saves". Obviously, your save files are supposed to be ending up in that folder.
Can you see if the game is saving files at all while the game is still running?