I had eczema and dandruff too, for 3 years now. Since I went Paleo, started supplementing with lots of vitamins after reading Dr Kruse's blog (including PQQ, Q10, E Tocotrienols, krill oil, among other multi-vitamins, and D3+K2+Ca+Mg -- I don't take all these every day, of course), started eating offal, cook with coconut oil, eat probiotic lactose-free goat yoghurt/kefir, raw & unfiltered LOCAL honey, all the dandruff went away as of magic within a few weeks.

Eczema persisted a bit longer, but it's already out after 3 months on the diet/protocol. I use the Rx Elidel cream on my forehead and ears, but now I only put some on 2-3 times a week only (used to be twice a day). I have no flare, I just use it as a precaution. I also bought some cheap face cleansing, and a day cream (nothing "natural", just the normal chemical stuff they sell for cheap). Still no flare.

Regarding shampoo, conditioner and toothpaste, I changed everything I used to use last month, to this brand: http://www.desertessence.com/ I'm not saying that they're the best, in fact my hair get less fluffy with their shampoo, but the point is, I made a major change, and the dandruff still didn't come back, which means that my dieting protocol "cured" the underlying problem that was creating these skin symptoms.

IF your skin problems are 100% skin-based and not the outcome of some internal problem as in my case, then you might have mucus problems, in which case you can try applying in your skin some castor oil, unrefined virgin coconut oil, and apple cider vinegar. Try each for 2 weeks, and then move to the next, to find which one works best for your case. And of course, you might need a combination of both the diet+supplement protocol I mentioned above, plus the external one.

Eat more fat! Especially coconut oil. Even take a bit and rub it on your skin and hair.
Also, it's good that you stopped using soap but you should also try to stop using moisturizer. Your skin may be dry for a while but it will compensate by increasing its own oil production. Just wean off it little by little. That's what I did. I used to use Aveeno moisturizer with SPF and it just made my skin worse. Now I just wash my face with water and if I do get a pimple, I use that Proactiv sulfur mask. It really is a miracle.
Another great fat is almond oil. Again, just use a tiny bit and it will make your skin so smooth.

In another question today, you mention that you are hypothyroid - might this not be behind the dryness?

Coconut oil really helps my skin stay well moisturized. I am naturally really dry no matter how much fat I eat. I exfoliate with sugar once a week, and use only coconut oil on my skin, and drink plenty of water. I don't see in your post anything about your water consumption. I've found that the vast majority of people are chronically dehydrated. Aim to drink half your body weight in ounces of pure water as a daily minimum. If you feel thirtsy, you are already dehydrated.

Put a humidifier in your room when you sleep? Also, I kind of heart argan oil as a face moisturizer.

Try salt water. Just dissolve some sea salt in water and rub the liquid on your skin a couple of times a day. This has helped me enormously. It can sting a little at first, but your skin will get used to it.

I've had very dry skin and atopic dermatitis since I was born, and no dietary intervention has ever gotten rid of the dryness. My skin is less itchy when I don't eat sugar, but still all flaky and dry. And it stays that way, no matter how much fish oil, coconut oil, or (insert your magic fat here) I eat. I have no idea why the salt helps, but it does. Try it now!

Iodine (Iosol, 4 drops daily) supplementation lessened my dry skin significantly. Biotin also makes my skin extra oily. Might try those.

Hi

Krill oil is great for the skin, check out this article, there's a lot of information on the topic. Plus lots more information on other ways to help your body.

http://revealgreatskin.com/krill-oil/

At the end of the day it's a combination of several different methods. Moisturise, eat right, take the good supplements and try not to let it get out of control.

Tried to cut out the minty toothpaste if you still use it? There are some threads about it and its relations to dry skin on the forum.

I have suffered from extremely dry skin since I was a teenager and only recently have i managed to overcome it. I apply an anti-wrinkle night cream, let it absorb then apply a beauty balm on top of this. I use products by mama nature but the principle would be the same no matter what products you used. I do this twice a week and apply the night cream only the remainder of the days. I use a wrinkle day cream by liz earle during the day. My skin is not dry at all any more and is fabulous to boot!!!!!

Is your dryness concentrated in particular locations, such as between the eyebrows, in the eyebrows, next to the nose? Is there associated redness and itching, or is it just flaking skin? What about the scalp...itching, redness, or just dandruff?

Edited: asking these clarifying questions to offer advice. Hopefully this isn't construed as a comment.