J.C.
I have a 17 month old and he does the same thing to me. I find distraction with toys, books or snacks help. I usually let him come with me when I go anywhere but there are times when I have to run something to the trash where I hear the ear piercing scream from the house when I walk away.
Ride it out, I went thru the same thing with my daughter and around 24 months it was gone and she only wanted her dad. Eventually it will stop and they won't need us as much anymore and we might miss our babies wanting us.