For basically anything rust-related, I use Evaporust. Usually, for chemicals, the safety to efficacy relationship is inverse; the more effective a chemical is, the less safe it is. Evaporust seems to have hit a nice sweet spot in that curve, where it is very safe, and very effective....mostly.
I learned the hard way that you should not use Evaporust on watch movements. Or at the very least, do not use them on any part of the going train if they're composed of steel. According to their website, Evaporust causes a phenomenon called "carbon migration." Since steel is Carbon and Iron, and Evaporust removes Iron, you're left with just carbon on the surface, including in between pinion leaves, which seemingly can only be removed via manual scraping. And you have to put a good amount of elbow grease into scraping it off, and one slip and you'll definitely destroy the part.
One watch was completely locked up, the other is limping by with maybe 60 degrees of amplitude.
Just putting this out there in case anyone else thinks to try it.