I have not found whiting to work well for me, but I've only used it several times. I make the paste, put it on, and it does not change color or seem to soak any oil out. I leave it on for several days and see no change. Maybe it only works on heavily oil soaked stocks and does not work as well on minor oil problems which seem to be recent issues.
Even with prolonged soaking in acetone I have found that oil will still come to the surface many days/weeks later.
I like to use heat at first to drive oil to the surface and to wipe as much away as possible, but at some point my wiping away seems to just push the remainder back into the surface of the stock.
So, the next step is to use ammonia or some other degreaser (purple power) mixed with very hot water. I just run it over the stock and then use a toothbrush to scrub the areas repeatedly. This seems to get the surface oil that I can't wipe off out of the stock.
Then I let the stock dry for a few days and heat again to see if oil continues to come out.
I try to stay away from a full acetone soak if at all possible as it seems that it really just removes too many good oils from the wood. I'll use acetone in spot treatments, but not a full soak.
Either way, acetone or water/degreaser, dries out the stock and I have to watch for small cracks developing that need attention before I start refinishing.