It would appear the problem is likely not on my end. I called my provider, and the guy I spoke to suggested trying it on a different device first.
I have a cheap laptop I picked up to help run the CNC router, and the guy I got it from had wiped it, so I did a fresh install for T-Bird.
It does the exact same things- downloaded the first four emails, then just gives me a copy of that same sales 'mail each time after that.
I called my provider back and told them about it, and they're supposed to be looking into it from their end. As of this moment- almost midnight on Tuesday- if I try checking my email again, all I get is a single copy of that same email again.
My Whiteboard email works, and the PayPal addy for Docsmachine works, so it's not, like, site-wide.
I'm hoping it'll be solved in the morning, but we'll see. I'm in the middle of a couple bits of rather important correspondence at the moment, so I'm sure not happy about it, but I guess ya gotta deal with it as it comes.