First of all, I think you misunderstood your doctor, because eggs are not dairy and are not related to milk protein. You may have an allergy to eggs, but it is separate from your allergy to milk.
Secondly, any reactions to alternative milks have nothing to do with your milk allergy. You may have allergies to almonds and coconut, or it may not be a reaction. But the only way to know is to see your doctor for allergy testing. What do you mean by making you feel sick? Are you having allergic reaction symptoms (hives, swelling, etc.?) or just not feeling well?
Have you truly eliminated milk from your diet? Could you be reacting to items with milk in them- baked foods, creams, yogurts, etc.? Milk shows up in some places you wouldn't expect. Look for casein and whey on labels; these are milk proteins.