In searching for a solution to my problem, I found this post from last November:
This is my problem exactly. But the solution isn't included in the post, and when I tried using the force option as suggested by @mvg ...
curl -XDELETE 'http://myServer/_watcher/watch/myWatch?force'
... I got the same illegal_state_exception / not started error.
So, @mcku, if you could let me know what your solution was, or if others can tell me how to fix this, I'd appreciate it. I saw that you mentioned being careful with trigger intervals -- I don't imagine that that's my problem, as this was running for months before this problem popped up (none of my watches are more frequent than once per minute). The server did run out of space recently, and we had to do a bunch of cleanup to get things running again, so I'm wondering if something related to that caused this problem. Some of the watch executions seem to be happening at irregular intervals sometimes as well, so that might be related. Thanks.