I have a 5 yr old daughter who is highly allergic to dairy. We never had any eczema problems with her or any symptoms of a dairy allergy when I was breastfeeding her. But, I suppose anything is possible.
If you're going to try to do a dairy free diet, here are some things to look for on labels.
First, anything with any obvious dairy (milk of any type, cream, butter, cheese, etc.), stay away from. You'll also need to look for dairy proteins such as whey and casine (sp?) in any form (including casinate). These are in everything!
Just recently, Jewel brand bread is now dairy free. But, since recipes and ingredients change often, you'll need to check the labels everytime. For a butter alternative, we use Soy Garden. It's completely dairy free and you can cook and bake with it (some soy butters you can't). Soy milk is actually quite tasty but we get the vanilla type. If you shop at Costco, you can get a 3-pack of 1/2 gallon size for pretty cheap. They also have a large tub of soy butter -- not Soy Garden but something very similiar (which is actually what we use now).
For our daughter, we also have to stay away from anything that was manufactured on equipment that also processes dairy. You probably don't have to worry about this so much right now. Lunch meat from the deli we never give our daughter because of the possibiliy of someone using the meat slider for cheese. Cross contamination is a big concern of ours. Packaged lunch meats are almost always okay but check the labels first.
I don't shop at Whoel Foods -- usually Jewel, Dominicks, and Costco. We get soy cheese at Fruitful Yield up by Fox Valley mall. I won't touch the stuff (smells horrible) but my daughter seems to like it. Then again, she's never had anythign but that. The downside (aside from the smell) is that it doesn't melt very nicely so cooking with it is a bit of a challenge. As far as rice cheese, veggie cheese, and soy cheese -- almost all of them have casinate in it -- a milk protein. So you have to be really careful. We've found the Tofutti brand to be fine. Their ice cream is actually really good too. I can't tell the difference.
I doubt at this point you have to go 110% dairy free but I hope I've given you some options.
Good luck. I hope you don't have a dairy allergy, or any other food allergy on your hands. :(
Jen