Oh, how do I dye thee? Let me count the ways.
Well, plus the above, you could make a "tea" out of yellow onion peels, then dunk it in there.
Or, you could use Potassium Permanganate, and dunk it until you get the right color.
You can also strengthen or weaken the solution by adding more pot perm to the mix or adding more water.
After I have aged the horn, I always put a coat of MinWax on it and polish with a soft cloth.
And more. That's all that came to mind right off hand.
BUT
What I would do first is to deglaze the horn with 0000 steel wool and get all the polish off. Might be a bit tricky with the engraving on it. Also, I would clean up the strap groove a bit more. 400 grit sandpaper wrapped around a piece of 1/4" dowel should do the trick. On the photo, I do see some "feathers" around the body side of the groove.
Have fun,
John (Bigsmoke)