Hi there,
Separation anxiety is listed at my pediatrician's office as a milestone. I think they do that so everyone knows that EVERY child will go through it at some point, or most at least.
You are breastfeeding because it's the best possible nutrition for your baby. Don't lose sight of that.
I bfed all three of mine, and only one was super clingy. I had to let dad take turns while I left the house. He cried, and that was ok, because his dad came to realize that he HAD to spend more time trying to bond with him. Once they spend more time together it will be better, so start small. Go run and errand or two. You will be grocery shopping by yourself in no time! Or even better, take your four year old to a movie! Time he sees as "mom and me" time, yet time to sit and chill.
You're a great mom! Keep that in mind! We all go through times like this, and it's really hard. Dads may not understand, my husband STILL thinks I did something wrong that made our now 2 year old so clingy at that age, but i did nothing different with any of them, and my 8 month old baby prefers HIM most of the time, lol! They are born with a personality, I tell ya! But she is too young to "train." Just be a mom, you'll do fine.
And remember there's support here when you need it. PM me any time, if you'd like!