Oh boy this sounds familiar . . . my MIL can take worrying to a whole new level. I can't imagine if she had something that severe to deal with.
My older teen had an emergency appendectomy on Thursday and we couldn't even mention it until it was all over.
I have tried to tell her (in a nice way) that the excessive worrying tends to drive people away because it is exhausting for the person on the receiving end of her anxiety. We all need people to vent to but when it's a one way street it's a problem.
I would try to let her know that you're going to have to talk with her less if she can't get a grip on her anxiety. You've got a young family to raise and you don't need additional anxiety in your life. If there is something concrete that you can do for your SIL then let her know you will do it, otherwise she needs to find another place to express most of her feelings. You are not her therapist nor are you her mom. SHE IS YOUR "MOM." Good moms don't dump on their kids (I have to remind myself of this when I start griping to my kids).
Having been there and done that, I don't hold out much hope for you guys. She probably won't heed your words, and you will have to cut down contact for your own sanity. You don't want to develop hard feelings toward her, and that's what I would tell my husband.
I'm so sorry, and I'm really sorry for your SIL. Bless her heart . . . she doesn't need this either. What she needs is strong, consistent, constructive and positive support.
Good luck.