I'd recommend that you do eliminate all 8 of the major allergens at one time as Sears suggests, plus any other foods you suspect as causes. And, I'd give it more than just a couple of weeks. Eczema can take more than that to heal, and it can take longer than that for the allergens to get out of the body.
My daughter is allergic to dairy, eggs, wheat, and a bunch of fruits. Eliminating just one at a time didn't fix her symptoms - although we started with eliminating just wheat and saw a noticeable improvement briefly. By eliminating all of them, you have a better chance of hitting on the one(s) causing the problem and you will shorten the overall length of the process.
Also, you may want to consider getting your pediatrician to run some blood test and/or going to the allergist for a skin-prick test. Neither one is as horribly painful as I was led to believe - Julie hardly even noticed when they poked her back! These tests are nowhere near 100% accurate, but they can give you good starting points on what you may need to trial eliminate.
Oh, and I'm editing to add that I just saw your fears about what to cook...I was right there just a few months ago, and I still feel frustrated sometimes, but once you get through the initial fear of it and have tried a few things, it gets a lot easier. There are great wheat-free noodles available that taste almost, if not just as good as wheat noodles. I usually still cook regular pasta for my husband and myself and cook a smaller portion of special noodles for her. We eat a lot of stir-fries and casseroles that involve rice or other grains like quinoa and oatmeal.
In fact, Julie eats fruit and oatmeal for breakfast almost every day, or Vans brand of waffles, which I sometimes eat too on days when I'm feeding them to her. We also buy wheat-free bread from the freezer section at the grocery store, which she accepts happily as a substitute. There are a couple of brands of crackers she can eat, and lately has been going nuts for hummus on them.
Common dinners I make include: tacos, spaghetti, stirfry with rice noodles like pad thai or the tiny vermicelli ones, pepper steak, curries, she loves this recipe for curry turkey burgers, which we enjoy too: http://www.foodnetwork.com/food/reci..._19830,00.html
and this recipe for portuguese kale soup: http://www.foodnetwork.com/food/reci..._29453,00.html
I make huge batches of that one and freeze it in individual servings for lunches or meals when I hadn't planned ahead well. I usually fish out the floaty stuff and put it on a plate so she can pick it up and eat it, then spoon-feed her the broth. Tonight we had beef stroganoff, she just didn't get the sauce on her portion. I'm sure I could think of some other ideas.
Anyway, you can do it - it just takes some planning and getting used to. There are a lot of allergy cookbooks available - just search for allergies in your local library's catalog.