Hope this is not too late.
Definitely NOT nail polish remover. There are a lot of chemicals there you don't want on your coins. As slokind said, get pure acetone from a hardware store, found on the paint dept.
And yes, be very careful with it. It is ridiculously highly flammable, it is bad to inhale it, it is bad to touch it, it is really bad if it gets in your eyes, etc. It will melt away latex gloves if you use them, it will destroy the paint on your desk or table if you spill it, it will melt your tennis shoes. Don't smoke when you use it, don't be near a heat source. I know it's a little exaggerated, I know you're not stupid, and I know you could have guessed all of this or read it on the can, but it's a lot better to be safe.
All that being said, this is what I do with the acetone. Bear in mind that this is what works for me, and there might be other methods that
work better, but I don't know about them.
Pour some acetone on a
glass container that has a lid (usually plastic, so don't let it touch the acetone; better, of course, if you don't have a plastic lid).
Get Q-tips (try to get that brand specifically, because they don't have plastic sticks that would melt away). Dip the q-tip on the acetone you poured and replace the lid (it will evaporate rapidly, and you don't want to inhale that stuff); hold your coin with another q-tip, and wipe lightly. You'll see the cotton turn green immediately. Don't forget to wipe around the edges.
Do on both sides and repeat, changing the q-tip every time until the cotton is white. You should now see a very different coin. Further cleaning might be necessary, but the goo/scuzz should be gone from the visible surface.
However, I would recommend letting the coin soak in the container overnight. There might be some goo/scuzz in the tiny crevices which the acetone doesn't reach. The next day you might see that the acetone has aqcuired a greenish tint. Take the coin out and let the acetone evaporate from the coin. Dispose of the acetone properly (refer to the container), but not on the sink because it might melt the
pvc plumbing.
I don't know what chemicals might be left on the coin, if any, since the acetone evaporates from the coin almost immediately, but to be on the safe
side, I let the coin soak for a few days in distilled water, changing the water every day. The coin might turn dull in appearance. Determine whether you want to clean the coin further (not the goo/scuzz, which should now be gone, but from other dirt or sediments or rust or whatever that's
still there). Either way, when you believe that the coin needs no further cleaning, use micro-crystaline Renaissance wax (or equivalent) to restore the shine.
Try this method on one coin first and see if you like the results before you do other coins.
I would post before/after pictures to illustrate, but I don't see a way of adding an attachment on this
thread.
Good luck.
