Ok, I think I've caused some confusion here. Some manual editing needs to be done between the GET and the PUT if you're creating a new index and needing to copy over the settings. The error message is identifying pieces of the payload that you'll want to just remove from the PUT. But then got an index already exists error because you used the index creation API for the approach where we don't need to create the index (it exists already), but only need to modify the mappings of an existing index. For that, use the PUT <index>/_mapping API you were trying originally. No settings necessary for that. Only need to worry about setting if you try to replace the original index.
// Delete the original, which was associated with the crawler
DELETE /search-sarah-test-5// Replace it with the new one. Error
What did you do for the "replace it with the new one" part? Given that the error complains that search-sarah-test-5 does not exist, I'm worried that the "replace" didn't work. Again, I think it's probably going to be more straightforward to edit the mappings, so maybe we should just shelve this "replace the index" approach for now.
I've now got a bunch of phantom crawlers that I can't delete
Oof, we're just hitting all the issues. I'm sorry for the experience you're having. This is a known issue we're working to get fixed. The workaround is described here: Ghost Web Crawlers - #3 by Jedr_Blaszyk