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Can scratches like these be removed or do you have to refinish? Do products like Old English scratch cover work on stocks. What about linseed oil?

Those scratches look deep enough that a surface conditioner won't do the job. I wouldn't worry about them.

I agree with Bob. Those look deep enough that a liquid scratch remover will not work. you could try using the ironing method.
Last resort is refinishing ikt. But if it's a user and not a safe queen, then I wouldn't worry too much about it, as it will most likely happen again.

I see the nicks as evidence the piece was used as it was intended. Think of them as been there, done that marks. It is a nice looking rifle.