[Arran]

Joseph;
What does the cabinet look like now? I would not be embarrassed about posting a picture of the after, the before certainly wasn't presentable, and the top veneer isn't a rare or exotic wood or anything. I have heard of people mixing oxalic acid into a paste an applying it directly onto stains, but what I am wondering is what those stains are? Are they actually water marks or are they something else like battery acid, or engine oil, or what? I'm wondering if you might try different solvents on the spots, see if they react?
I once got some black marks out of walnut by using a phosphoric acid based rust remover. Maybe try wood bleach, or that stuff they use to revive wood decks, on small areas of course? You really have nothing to loose. If all else fails you could paint the cabinet, or use extra dark walnut tone spray all over the bad parts.
Regards
Arram