I try to stop at an inconspicuous spot, like in a ditch, or on a patch with a busy print stopping on the exact color of the thread. As for the extra on the bobbin, I will keep it with all my other partially filled bobbins and use it later either in the bobbin, or as the top thread when I am machine basting or working on a small project.
Now when all bobbins are "full", and I need a new bobbin to fill, I choose the bobbin with the least amount of thread, and/or least likely to use, pull the thread off it and trash the unwound thread. That is what I do...hope it helps you to decide what to do. Jane