I'm lactose intolerant but when I drink 1 or more glasses of lactaid I feel sick?

Kiara • Hi! My name is Kiara.
I tried drinking lactaid but it gave me cramps, and made me sick. I looked it up and I found that maybe I am allergic to the proteins in milk. But the thing is, the doctors confirmed that I'm lactose intollerant. So I don't know if the milk protein allergy is a possibility?