It might be a bit late now, but I try to outline that expectation upfront with my FWB. I add it to my boundaries. Something along the lines of "no emotional or personal burdens from either of us". If they try, I usually gently remind them with something along the lines of "I'm really sorry you are going through that. I hope you have someone that you can talk to about it and who can emotionally support you with that".
I've got a demanding job, marriage, kids, etc and my own stuff to worry about. Adding the emotional burden of another person just takes the fun out of having a FWB relationship. Don't get me wrong, I like chatting with and connecting with people, but I like to keep it lighthearted and fun.
I'm sorry I don't have any specific advice for you right now since you seem to be in the thick of it. But I would try and outline that expectation upfront next time.