whiting and acetone is good for removing spot stains. I would remove cosmoline and oil too, I assume.
I prefer oven cleaner, rinse in HOT almost boiling water, oven cleaner again, rinse in hot water.
I have never had a stock, walnut or otherwise, discolor or look "burnt".
If I have a special stock that I want to be careful with, then I use denatured alcohol and heat. I clean with denatured alcohol (at Walmart by the paint strippers). Then I steam the stock with a wet rag and a cheap clothes iron.
I cannot convey to you how much much oil and cosmoline will come out of the stock by heat alone. It is mystifying. The oven cleaner is VERY fast, but only get some of the oil out. It gets out enough to handle the rifle with having to wash hands or change clothes, but the summer sun always brings more out.
If in doubt, then do all three methods: oven cleaner, steaming, and baking in the sun. There are plans online for "cosmoline cooker" too.
In the winter I have stood a rifle shipping box on end, placed the stock inside, and pushed the heat gun into the end of the box. When it started to smell, then I took the stock out and wiped it down. Put it back in the box, heated, and wiped. OBVIOUSLY YOU CANNOT LEAVE THIS UNATTENDED.
It is like sanding. If you think that you are done, then you are not. There is always more cosmoline and oil that will come out of the stock.