Crafting exceptional (or remarkable/legendary) items is based on a few things:
Firstly, the quality of the resources used. So for example, if you use exceptional items in the craft, you have a higher chance at crafting higher quality items. The maximum bonus you can get is if all items used in the craft were Exceptional (or Legendary).
The second factor is your skill in Stonecrafting (or any other applicable skill in which ever craft you are doing). The higher, the more quality bonus chance you get.
Another easy way to get rid of items is to throw or drop them in deep water. Be careful not to drop anything important!
There’s also an option to “Drop on Gather” if you are playing the weekly version which will allow you to drop any items you get on gathering.