I have the same thing with meat, eggs, and cheese.
There are many egg substitutes out there if you need an egg in something you are baking, just Google it. No really there are thousands of subs you can use. As for the milk you can try rice/almond/soy milk or other milk alternatives, same with the cheese. For grains you can get your fiber from many vegetables such as broccoli, beans, cabbage, etc.
Check with your doctor to see if you are lactose intolerant. Also, though I haven't been diagnosed, it was suggested by a friend that I may have psychophysical disorder, which you might also want to look into.