Re: collapsing picket fence.
If the drawing is accurate, it's interesting that you're losing a post that is in line with the fence because the rest of the fence should be supporting it vertically.
If the slope is stable, then installing a longer post to give it more purchase should do the trick. When you replace the post, reset the fence to it and get everything where it needs to be before setting the concrete to it. You may want to install a cross brace on the post to help keep it vertical in the future.
I suffer from CDO ... Its like OCD, but in alphabetical order, LIKE IT SHOULD BE!!!