Milk Allergy?

Does anyone baby have a milk allergy? I’m waiting for his appointment to be absolutely sure, but he is constantly grunting, and gassy. He literally screams when he’s pooping or farting and his poop and farts smell like rotten eggs. He’s started to cough after he eats, and he has diarrhea every poop. He also arches his back and draws his legs up when pooping or farting. I switched him to a hypoallergenic formula today, so I’m waiting to see if it works, I was just wondering if this for sure sounded like an allergy.