I think I have a solution!
My hair is the same way - thick and dark (very noticable on a pale girl)... Shaving never works for me. The roots of the hairs are still visible, and I get stubble incredibly quickly! Nair isn't the answer either.. you get the same results you get from shaving. Shaving = cutting the hairs, Nair = chemically dissolving hairs (yet leaving behind the roots, like shaving).
So what is the answer, you ask? WAXING! There will be no darkness on your underarms because the hairs are removed completely - roots and all!
If you're willing to try waxing, I recommend Nair's microwavable wax. The cold wax never worked for me. Oh, and the pain isn't bad.. It's hardest the first time; waxing is easier every time after! But the best thing about waxing is the hair doesn't grow back for about a month... I hated having to shave every day.
Hope you give it a try!