I hope you have luck cleaning up the Lie-Nielsen saw. I feel your pain: I'd hate for such a fine tool to suffer the indignity.
Once you get it tamed, it's good to turn to prevention. How about camellia oil? So simple. I use the tried and true method of a sized and neatly folded microfiber cloth slightly overstuffed into an Altoids can. You can bend the lid to close tight even over the stuffed towel. I spray the camellia oil liberally over the top surface of the cloth. When I pop the top of the tin, the cloth puffs proud a bit. Then I just glide the open can along the saw, both sides, and maybe finish up with a gentle wipe with a clean paper towel to moderate the oil layer. The saw gets put away oil-resistant. The can gets popped closed.
I do the same thing with plane irons after honing, hand plane bodies after use, chisels, Forstner bits, router bits, and my marking knife –– any steel that could rust on a hand tool or small cutter. Tables on machines get Boeshield if I'm worried about rust.
I'm probably just dumb lucky living in California where rust is less of a concern, but even at the end of a damp winter I have no rust.