Hi Lin,
Nice find!
Don't soak the corks. If they're fresh and dry from the store, insert them fresh and dry. I used to soak my corks in a K-Meta / water solution, just dipping the corks in it before inserting them into the bottles. When I did this, my corker got wet inside the working mechanism, and parts began to gather crud and stick, maybe even rusting a little inside. The corker started shredding a little cork off the cork and inserting cork debris in the bottle as it corked. I'm about to tear it apart or replace it now. It was pointed out to me by SHoffner, while we were bottling, that the corks are pre-sanitized and pre-lubed from the manufacturer, no need to soak.
As for cleaning, I'd just scrub it top to bottom, maybe with a product like PBW, then rinse thoroughly, then sanitize. Let it dry and try corking a few bottles, see how it works. Or do a test run on a couple (empty) bottles just like it is to see how it's working. Worst case scenario, it may need taken apart and cleaned inside, and lubed with a food grade lubricant. But maybe just a good cleaning, sanitizing and drying is all it will need.
You're getting good at finding winemaking deals!