Old scales

By the washers and grain I would say they are horn.
I have a William Gilchrist with similar scales.Attachment 353672

Definitely horn. Old, dry horn often looks woody. Lots of people make that mistake.

If the blade closes fine, just go very lightly with the grain (length) with very fine steel wool to clean. mash a bit down flat and go in between the scales as you can
Lightly oil with mineral or mink oil wiping on gently with a soft bit of cloth running it in between the scales as well.
A little at a time, wait a day before doing it again. Nice results will come within a few applications.
Too much oil will often soften the horn too much or cause warpage and cannot be reversed.