They are the bane of my existence--I have all the characteristics your derm described, very fair skin with a bluish undertone, so they just leap out at you. The skin of the face is thinnest under the eyes, and if you are already pale you probaby have fairly thin skin, so the blood vessels really stand out. I already have the kind of bone structure that shows them off, high cheekbones, thin-ish cheeks,and really deepset eyes (this tends to get more pronounced with age, I've noticed--yay!)--and when I've not had much sleep, stayed up drinking red wine and eating salty things, I look like Keith Richards the day after.
Back when I was in design school, I was working a full graveyard and then going to class all day, and I'd have to give regular presentations that might be attended by future employers so I'd have to look halfway presentable, and you'll laugh at what I used: Preparation H. It sounds crazy, but if you think about it, it works by shrinking inflamed blood vessels, and that's what causes those horrid dark circles.
I'd put it under my eyes in the morning--yeah, it's gloppy and sticky and you probably don't want to get it in your eyes, slather the rest of my face with Clinique Moisture Surge, then when that was absorbed, I'd slap Clinique Total Turnaround cream over it (tightens the skin and leaves it feeling nice and soft), and pound water all day. Then just before I had to go out there, put Estee Lauder Spotlight cream on--it has really soft rosy gold light-reflecting particles in it,probably like that Prescriptives mineral stuff, subtle shimmer, not disco glitter.
Result, even if I felt like death, I did not look like I was about to start singing Paint it Black, and if it worked for me, it would probably work for anyone.
No, I do not work for Estee Lauder or Clinique, by the way!