I wonder???? Marks of use may be a good thing if you're going to keep it. For instance, blood on the sling, scratches on the stock where it was held against a tree to make the shot on that 8 point and such things give a lot of character.
To attempt to answer your question. If its an oil finish and the "dirty" spots are in the checkering area it would seem that the grain end cuts allowed body oil to soak in. If so, recheckering may be in order.
I know of no material/solution that could be used that most probably not cause other color mismatches.:confused:
AKA - the guy who knows the guy who
will be heralded as the man who kept elk from extinction.