RE: Your horrifying news....
Jamie -
Looking at those little chunks, they look a lot to me like tub or centerpost rust.
I don't know know long your machine sat unused, perhaps not at all, but there is often rust and debris build-up on centerposts and on both inner and outer tub surfaces. I agree with Steve/Gyrafoam that your issue warrants a basket removal. You'll know for sure what you're dealing with.
If the machine went unused long enough to fully dry out, my experience has been that build-up shrinks and becomes loose, which would then get dislodged when the machine is put back into service. If those rust chunks did indeed dislodge from somewhere, and then after that you had oily marks on clothes, my guess would be that either the agitator shaft seal is in need of replacement, or there's a hole in the center post. The rust could also have come from the basket itself, but that is not as likely.
Agitator shaft seals are commonly in need of replacement (lets say always if they haven't been done in a while or ever), and when they're replaced, they help to keep the centerpost dry, and thus free from further damage. This little cheap seal is what helps the air bubble to form under the agitator. When the seal fails, the air bubble doesn't form, and water goes up the centerpost, causing it to rust eventually and water to find a pathway into the mechanics of the machine. I can't tell you how many machines I have seen that have "chocolate milk" looking oil/water in their transmissions from centerpost leakage.
If you can, I strongly recommend pulling the tub to see what you have going on. If you can't do that, an easy way to tell is to pull the agitator after you've just done a wash load. The agitator from a well sealed machine will be mostly dry underneath from about three inches from the bottom, up the center. It's normal for the base to be wet obviously, but if you see water beads or droplets way up the center of the agitator toward the splines, that is an indication that seals aren't working and is probably how the machine exchanged some oil.
In many cases I have seen agitators and baskets that had white detergent residue on them in a water-mark line from when the machine was sealing properly. If yours has this too and is wet well above, that's another indication of a seal failure.
I don't want to alarm you, and this does not mean the end of the machine for sure, but I would check this before I use the machine much further just in case.