You’re trying to run PoE, which is a 64-bit game, on a 32-bit OS. It just won’t work.
If your CPU does support 64-bit instructions and you don’t want to go through an OS reinstallation to switch to 64-bit (despite the fact it would be a good idea), just tell me and I’ll guide you through the process of enabling 64-bit applications support on your Ubuntu.
If you don’t know if your CPU does support 64-bit instructions, post its model here and I’ll try to guess if it does. Anyway, if you bought it in the last five years it probably does support 64-bit instructions.
It may be the running as root that disabled the ability of your user to save games.
Have a look on this thread (GOG forums):
The directories you should check are listed on this post: