Thanks for trying to help, but to be honest deleting 15-20 separate accounts and recreating them will be a colossal amount of work - plus anytime I have ever deleted and re-added a mail account, it brings in all the same past tons of emails off the server. I do not want to dump my servers - these are work servers with vital historic emails I purposely keep there in case we ever need to find them again.
BTW, now one account has a bunch of emails dated 12/31/69 - LOL!!!!!
UPDATE:
This is the strangest thing I've ever encountered. Here's a new twist: I have found that if FIRST I manually swipe one or two emails to manually delete, THEN it will let me "delete all" and then I have to also manually swipe/delete a couple of the individual emails from inside Trash (to dump them from Trash) and THEN it will let me dump the trash.
Still it then pulls all the previous 250 emails sitting the server (so now I'm getting ones dated July 2015 and previous) and have to do it again, but then it seems to stop bringing them in.
I don't know if this will hold - but whatever has changed its something that's caused the iPhone mail to pull in 250 emails from their respective servers.
I hope that this will work for good and hopefully it will help anyone else having this oddity!