PLEASE do not get a tan from the sun or even worse a tanning bed. It wil RUIN you skin and even worse leave you more suseptable to skin cancer (I know thats what everyone else has told you but it cannot be stressed enough)
I think a fake tan may help as it would bridge the gap between pale skin and dark circles. I like Fake Bake for the face, Olay Daily sun, Clarins facial fake tans but there are tons on the market. I just named these as I am very pale too. I would then recommend Bobbi Browns under eye colour corrector. Its awesome for cutting the dark shadows, I also love Amazing cosmetics concealer, its apparently got the highest amount of pigment in any concealer, Anyway, they both work for me. Good Luck