S.
L.,
I have an almost 3 year-old who does the same thing. What I do with her is tell her in a firm voice that I will not listen to someone who is whining. If she wants something, she must use a big girl voice. Also, when I've said no to her for something and she continues to whine and ask for it, I tell her I've already answered her question, and I'm not going to answer it again.
I've found that when she gets into a whining loop (can't seem to stop), that I go to my "happy place" and become a robot! I get done what needs to be done and when finished with her, I remove myself or her from the situation. I tell her that when she is ready to talk to me like a big girl, then she can join me.
Hope this helps. It sounds like you have a lot on your plate. Do you have family or good friends around to help? It sounds like you need some time to yourself.
S.