So I ended up pulling out the old BB and this is what the BB Shell looked like on the frame.

I cleaned it up a little with a small bit of paint thinner and then re-greased it.
Here is what my BB looked like after I did a little bit of cleaning. It's in really good shape and when I took it to my local bike shop, the mechanic said I didn't even need new bearings.
This is what the the 600ex BB looks like put together.
I ended up putting a new BB though (Shimano UN-55) because I had the tool to put in a cartridge sealed BB, but not the 600ex. They did not sell the tool to tighten it in and I couldn't wait a week to order one since I wanted to ride my bike this weekend. I did save the old BB though and will surely keep it around if the new one fails or if I want to put it into a new bike because it appears to still be in great condition even after 28 years.