Scrimshaw... well, that is a matter of definition. Basically the artwork on cow's horn isn't scrimshaw, though is similar. I've got some horns with beautiful scratching and maps including one I just rehabbed from the dog getting it when she was a pup. I added to the original scratching. (photos) Now looking for ideas to fill in the one blank space able to be seen when the horn is tipped to drink. Horn carving is entirely different. It is actual taking away of material from the horn surface like you would with wood. It is not scratching or scrimshaw.
To answer your question, I hear artists using all sorts of things to use to color in the lines. (some horns had color as well as black line) I personally just use a sharpie permanent marker and a rough old washcloth to wipe away the excess off the surface. I do that for the colors now too. It helps to
get the lines deep. Thanks to a friend making me as scriber with a thicker handle, I can do that now.
Biggest thing is to
Have Fun doing it. Don't let yourself get frustrated.
Horn after repair before new scratching:
Horn after new additions. Note the puppy tooth marks still visible
Top. added shading and a few 'rays'
Added to the basic moon, corrected the nose and added stars then addd the 'trip' around both sides. corrected best I could, the Lettering. the U lookedlike a V and the M an N with a short tail. I may add a 3rd vertical line at the front of the R to make it more 'attached' to the letter.
Blank placeoposite sipping opening. ... undecided what to put here.
there is an etching of mental heath and shows two drunks. I may do something like it.
Drawing is from an 18th-century-engraving-of-alcoholics-national-library-of-medicinescience-photo-library
a characture of the one on the Right might be what I do.