You use "out" when you've gone out for the day, or gone out shopping etc, "away" is used for things like away on holiday or away for a few days.

It's about time and distance. Out implies shorter time and shorter distance, away implies longer time and longer distance.

just so you know both of your examples are erroneous. 1. I'll be out next days. this could be corrected to I'll be out for the next few days. 2. I'll be away next days. This could be corrected to I'll be away for a few days. Sorry, i don't know why they're wrong, just that they're wrong. good luck with it all!