I recently decided I'd boot up Morrowind again after a long time of not playing. I knew that my previous installation had been HEAVILY modded and I wasn't entirely sure what was up there. I figured I'd boot up a (mostly) vanilla run of the game with nothing but visual enhancements (better bodies, MGE, and the graphics overhaul.) I'm having a problem, after installing the game and the expansions, I launched it and looked in the data files. And in them, was EVERY mod that I had on my previous installation. To my knowledge, there are no instances of the mod files anywhere on my computer, and not in my Morrowind Data Files folder. I booted up the game, and oddly enough, all of the meshes from Better Bodies and the heads mod I had installed (name escapes me) were loaded on Jiub, and the textures of the ship were obviously modified.

This is not only weird, but maddening trying to fix. Has anyone had similar issues and is there a quick fix?

Are you on Vista or Win7 ?
I had a similiar problem a few years ago.
The files may appear hidden if you don't have Admin permission over the folders (not the same as being signed in as Admin) but still be used by the game.

The easiest way to bypass all problems like that is to simply change the install location of Morrowind out of Program Files.
You can just copy/paste your Bethesda folder elsewhere.

Turning off your User Account Control (UAC) will also have the same effect.

Funny that you mention that, I booted the launcher in Admin mode and it completely fixed the problem. I'm still going to try and root out whatevers causing it though, it appears to be MGE and the overhaul are left there, so perhaps if I'm on uninstall in the control panel, I could flush it out.
Also, I'm currently running Windows 7.

The problem is specific to Vista and Windows 7. It's how they are designed.
It's not a mod problem, or a Morrowind problem.
The system itself has these "safeguards" that we all have to disable for many typs of programs that are installed in Program Files.